Question

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) = √x [0, 9]

c=?

I've tried quite a few different answers

Answer 1

f(x) = √x
f(0) = 0
f(9) = 3
So, the slope of the secant is 1/3

f'(x) = 1 / 2√x
So, we want to find c such that

1 / 2√c = 1/3
√c = 3/2
c = 9/4
So, c is in the interval [0,9]

just to check,
f(9/4) = 3/2

So, the equation of the tangent line at c is

y = 1/3 (x-9/4) + 3/2

http://www.wolframalpha.com/input/?i=plot+y%3D%E2%88%9Ax%2C+y+%3D+1%2F3+%28x-9%2F4%29+%2B+3%2F2+for+x%3D0..9