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For the Function ​ (A)Find the Critical Numbers of F

Question 90

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For the function For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ​ (a) Find the critical numbers of f (if any) ;
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema.


A) (a) For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(b) increasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; decreasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(c) relative max: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; relative min: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​

B) (a) For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(b) decreasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; increasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(c) relative min: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; relative max: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​

C) (a) For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(b) increasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; decreasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(c) relative max: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; relative min: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​

D) (a) For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(b) decreasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; increasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(c) relative min: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; relative max: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​

E) (a) For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(b) increasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; decreasing: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​
(c) relative max: For the function   ​ (a) Find the critical numbers of f (if any) ; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ A)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ B)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ C)  (a)    (b)  increasing:   ; decreasing:   (c)  relative max:   ; relative min:   ​ D)  (a)    (b)  decreasing:   ; increasing:   (c)  relative min:   ; relative max:   ​ E)  (a)    (b) increasing:   ; decreasing:   (c) relative max:   ; relative min: no relative min. ​ ; relative min: no relative min.

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