Can someone help me out with this problem by including steps for each? Find the

Question

Can someone help me out with this problem by including steps for each?

Find the x-value of all points where the function below has any relative extrema. Find the value(s) of any relative extrema. x2 5x-1 (x) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function has a relative maximum at the point(s (Simplify your answer. Type an ordered pair Use a comma to separate answers as needed.) O B. The function has no relative maximum. Select the correct choice below and, if necessary fill in the answer box to complete your choice. O A. The function has a relative minimum at the point(s (Simplify your answer. Type an ordered pair Use a comma to separate answers as needed.) O B. The function has no relative minimum.

Explanation / Answer

Solution :

To find the relative extrema , we need to find the critical points.

Critical points are those where the first derivative of the function is equal to zero .

f(x) = -x2 +5x -1

First derivative of the above function :

f'(x) = -2x + 5

f'(x) = 0

-2x + 5 = 0

-2x = -5

or x = 5/2 = 2.5

So x = 2.5 is the only critical point.

Now we will find the second derivative :

f''(x) = -2 <0

As the second derivative is smaller than zero , so by the second derivative test x = -2.5 is the point of relative maxima.

f(2.5) = - (2.5)2 +5(2.5) -1 = -6.25 + 12.5 -1 = 5.25

So the given function has a relative maximum at the point (2.5 , 5.25) .

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