Question

Given the function `f(x)=1/sqrt(x+2)`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-1,23] and find the c?

Answer 1

Answer the questions: Is `f` continuous on the closed interval? Is it differentiable on the open interval? Solve `f'(x) = (f(23)-f(-1))/(23-(-1))` in the open interval.

Explanation:

The function is continuous on ites domain `[-2,oo)`, so the first answer is, Yes.

`f'(x) = -1/sqrt(x+2)^3` which exists for all `x > -2`. So the second answer is, Yes.

The only solution to

`-1/sqrt(x+2)^3 = (1/5-1)/24`

is `x=-2+30^(2/3)` which is in the open interval. So that is `c`.