Answer:
See Explanation
Step-by-step explanation:
Your question is incomplete. I will take the complete question as:
[tex]f(x) = -x^2[/tex]
[tex]Interval: [-1,1][/tex]
Required
Determine whether Rollie's theorem can be applied or not
First, find the coordinates of the endpoints on the closed interval [-1, 1]
[tex]f(x) = -x^2[/tex]
[tex]f(-1) = -(-1)^2 = -1[/tex] ------ [tex](-1,-1)[/tex]
[tex]f(1) = -(1)^2 =-1[/tex] ---- [tex](1,-1)[/tex]
Find the derivative of f(x)
[tex]f(x) = -x^2[/tex]
[tex]f'(x) = -2x[/tex]
Calculate the slope (m) of the secant line i.e. [tex](-1,-1)[/tex] and [tex](1,-1)[/tex]
[tex]m = \frac{-1 - (-1)}{1 - (-1)}[/tex]
[tex]m = \frac{-1 +1}{1 +1}[/tex]
[tex]m = \frac{0}{2}[/tex]
[tex]m=0[/tex]
Set the derivative to the calculated slope:
[tex]f'(x) = m[/tex]
[tex]-2x = 0[/tex]
[tex]x=0[/tex]
Plug in 0 for x in f'(0)
[tex]f'(x) = -2x[/tex]
[tex]f'(0) = -2 * 0[/tex]
[tex]f'(0) = 0[/tex]
Since f'(0) is differentiable at x = 0. We can conclude that Rollie's theorem can be applied
