2.Answer the following questions about the function whose derivative is f'(x) (x-5) (x+8).a. What are the critical points of f?b. On what open intervals is f increasing or decreasing?c. At what points, if any, does f assume local maximum and minimum values?a. Find the critical points, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.A. The critical point(s) off is/are x5, 8(Simplify your answer. Use a comma to separate answers as needed.)O B. The function f has no critical points.b. Determine where f is increasing and decreasing. Select the correct choice below and fill in the answer box to complete your choice.(Type your answer in interval notation. Use a comma to separate answers as needed.)O A. The function is increasing on the open interval(s)and decreasing on the open interval(s)O B. The function f is increasing on the open interval(s)and never decreasingO C. The function f is decreasing on the open interval(s)and never increasingc. Determine the local maximum/maxima, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.O A. x=(Simplify your answer. Use a comma to separate answers as needed.)O B. There is no local maximum.Determine the local minimum/minima, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.O A. x=(Simplify your answer. Use a comma to separate answers as needed.)O B. There is no local minimum.

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Asked Oct 27, 2019
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2.
Answer the following questions about the function whose derivative is f'(x) (x-5) (x+8).
a. What are the critical points of f?
b. On what open intervals is f increasing or decreasing?
c. At what points, if any, does f assume local maximum and minimum values?
a. Find the critical points, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The critical point(s) off is/are x
5, 8
(Simplify your answer. Use a comma to separate answers as needed.)
O B. The function f has no critical points.
b. Determine where f is increasing and decreasing. Select the correct choice below and fill in the answer box to complete your choice.
(Type your answer in interval notation. Use a comma to separate answers as needed.)
O A. The function is increasing on the open interval(s)
and decreasing on the open interval(s)
O B. The function f is increasing on the open interval(s)
and never decreasing
O C. The function f is decreasing on the open interval(s)
and never increasing
c. Determine the local maximum/maxima, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. x=
(Simplify your answer. Use a comma to separate answers as needed.)
O B. There is no local maximum.
Determine the local minimum/minima, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. x=
(Simplify your answer. Use a comma to separate answers as needed.)
O B. There is no local minimum.
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2. Answer the following questions about the function whose derivative is f'(x) (x-5) (x+8). a. What are the critical points of f? b. On what open intervals is f increasing or decreasing? c. At what points, if any, does f assume local maximum and minimum values? a. Find the critical points, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The critical point(s) off is/are x 5, 8 (Simplify your answer. Use a comma to separate answers as needed.) O B. The function f has no critical points. b. Determine where f is increasing and decreasing. Select the correct choice below and fill in the answer box to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) O A. The function is increasing on the open interval(s) and decreasing on the open interval(s) O B. The function f is increasing on the open interval(s) and never decreasing O C. The function f is decreasing on the open interval(s) and never increasing c. Determine the local maximum/maxima, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. x= (Simplify your answer. Use a comma to separate answers as needed.) O B. There is no local maximum. Determine the local minimum/minima, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. x= (Simplify your answer. Use a comma to separate answers as needed.) O B. There is no local minimum.

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Expert Answer

Step 1
(a)
In order to find the critical point, put the derivative function equal to 0 and solve for x as
follows
f(x)0
(x-5)(x+8)=0
(x-5)0 or x +8 = 0
x 5 or x-8
Hence, the option (A) is correct. That is, the critical points of the function are 5 and -8
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(a) In order to find the critical point, put the derivative function equal to 0 and solve for x as follows f(x)0 (x-5)(x+8)=0 (x-5)0 or x +8 = 0 x 5 or x-8 Hence, the option (A) is correct. That is, the critical points of the function are 5 and -8

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Step 2
(b)
Note that, a function f is said to be increasing on an interval I, if its derivative is greater
than zero (f0) and that of decreasing, if its derivative is less than zero (f <0).
Hence, apply the increasing and decreasing conditions on the derivative of the function and
obtained the intervals as follows.
Increasing:f'(x)> 0
(x-5)'(x+8)>0
x8(x-5)0
Interval notation:(-8, 0
Decreasing: f'(x < 0
(x-5)(x+8)<0
x-8 (x-5)>0)
Interval notation:(-0,-8)
Hence, option (A) is correct
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(b) Note that, a function f is said to be increasing on an interval I, if its derivative is greater than zero (f0) and that of decreasing, if its derivative is less than zero (f <0). Hence, apply the increasing and decreasing conditions on the derivative of the function and obtained the intervals as follows. Increasing:f'(x)> 0 (x-5)'(x+8)>0 x8(x-5)0 Interval notation:(-8, 0 Decreasing: f'(x < 0 (x-5)(x+8)<0 x-8 (x-5)>0) Interval notation:(-0,-8) Hence, option (A) is correct

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