Question

verify that the fn satisfies the hypotheses of the Mean value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the theorem:
f(x)=3x^2+2x+5 [-1,1] i took the a.d. and got x^3+x^2+5x then plugged in the interval but didn't get 0 as the answer should be

Answer 1

correct me if im wrong, the mean value theorem states that:
\[\frac{f(b)-f(a)}{b-a} = f'(c) \]
for some c in the interval [a,b] correct?

Answer 2

yes that is the mean value theorem

Answer 3

What you want to do is calculate the slope using the endpoints of your interval:
\[\frac{f(1)-f(-1)}{1-(-1)} = \frac{10-6}{2} = 2\]
now you want to find out what number 'c' inside that interval gives f'(c) = 2, so set that anti derivative equal to 2:
\[x^3+x^2+5x = 2 \Rightarrow x^3+x^2+5x-2 = 0\]

Answer 4

wait...i just read what i posted...you dont want the anti derivative, you want to take the derivative of f(x).

Answer 5

you want f'(x), so the derivative would be:
\[f'(x) = 6x+2\]
and you want to set that equal to 2

Answer 6

so we get:
\[6x + 2 = 2 \Rightarrow 6x = 0 \Rightarrow x = 0\]
and because it is in our interval [-1,1], it is the correct answer.

Answer 7

ok so the derivative part is what i was doing wrong! thanks so much for you help