Rolles theorem on f(x) = Xsqrt(64-X^2) on the interval [-8,8]?

Question

The function f(x) = x(64-x^2)^1/2 satisfies the hypotheses of Rolle's Theorem on the interval [-8,8]. Find all values of that satisfy the conclusion of the theorem.

a.) + 1, AND -1

b.) + 4sqrt(2) and -4sqrt(2)

c.) 4sqrt(2)

d.) 1

My answer. The intervals do match and equal zero so Rolles theorem can work.

Second I found the derivative maybe thats where I can't solve this problem.

The derivative that I got was 64-x^2+x/sqrt(64-x^2) maybe i did wrong on the simplifying. I at least tried hopefully some one can explain as much as possible with every single step because I can figure out the algebra part.

Answer 1

your derivative should have been sqrt(64-x^2) - x^2/sqrt(64-x^2)

The minus sign coming from the derivative of (-x^2)

Setting the derivative to zero,

sqrt(64-x^2) - x^2/sqrt(64-x^2) = 0   multiply both terms by sqrt(64 - x^2)

(64 - x^2) - x^2 = 0

64 = 2x^2

32 = x^2

x = +/- 4.sqrt(2)

Answer: option b