Question

Verify the conditions for Rolle's Theorem for the function f(x)=x^2/(8x-15) on the interval [3,5] and find c in this interval such that f'(c)=0

I verified that f(a)=f(b) and calculated f'(x)= (8x^2 -30x)/64x^2 -240x +225)
But I'm having trouble finding c when that derivative is equal to 0.

Answer 1

I find it a bit easier not to expand the derivative

f'(x) = 2x(4x-15)/(8x-15)^2

Clearly f'=0 when x is 0 or 15/4

So, f'(15/4)=0, and 3 < 15/4 < 5

f'(0)=0 also, but 0 is not in [3,5]

The graph at

http://www.wolframalpha.com/input/?i=x^2%2F%288x-15%29+for+3+%3C%3D+x+%3C%3D+5

clearly shows that f'(3.75) is zero.