Question

 (a) Does the function f(x) = x 3 satisfy the hypothesis of the Mean Value Theorem on the interval [−10, 10]? Justify your answer. (b) If so, determine the point c which is quaranteed by the Mean Value Theorem. 

Answer 1

Given function, $f(x)=x^3$

(a) As the given function is a polynomial function, so It is differentiable and continuous, so by the mean value theorem, there exists a point in the interval [-10,10].

(b) The point c is given by

$f'(c)=\dfrac{f(b)-f(a)}{b-a}=\dfrac{f(10)-f(-10)}{10-(-10)}=\dfrac{(10)^3-(-10)^3}{20}=\dfrac{2000}{20}=100$

$3c^2=100\Rightarrow c=\sqrt{\dfrac{100}{3}}=\dfrac{10\sqrt{3}}{3}$