Question

Find the maximum or minimum of the function.

y = −x2 − 8x

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the maximum or minimum of the function y = -x^2 - 8x, we first need to find the critical points of the function.

To do this, we take the derivative of the function with respect to x:
y' = -2x - 8

Next, we set the derivative equal to 0 and solve for x to find the critical points:
-2x - 8 = 0
-2x = 8
x = -4

Now that we have found the critical point at x = -4, we can determine if it is a maximum or minimum by using the second derivative test.

Taking the second derivative of the function:
y'' = -2

Since the second derivative is negative, the critical point x = -4 is a maximum.

Therefore, the maximum of the function y = -x^2 - 8x occurs at x = -4.

To find the maximum y-value, substitute -4 back into the original function:
y = -(-4)^2 - 8*(-4)
y = -16 + 32
y = 16

Therefore, the maximum value of the function is 16.