Free Skill check: Mean value theorem, L'Hopital, and local linearization Practice Problems • Learn To Be

The Mean Value Theorem states that if a function

f

is continuous on

[a,b]

and differentiable on

(a,b)

, then there exists a real number

c\in(a,b)

such that

\qquad \displaystyle f\,^{\prime}(c) = \dfrac{f(b)-f(a)}{b-a}

.

Which of the following functions do NOT satisfy the conditions of the mean value theorem?

Choose all answers that apply:

A
$\text{I.}$ $\qquad f(x) = \left\{ \begin{array}{lr} \sin{x},~~~~~~0 \leq x \leq \dfrac{\pi}{2}\\ \cos{x},~~~~~\dfrac{\pi}{2} < x \leq \dfrac{2\pi}{3} \end{array} \right.$ $~~~~~\text{on}~~~~ \Big[0,\dfrac{2\pi}{3}\Big]$
B
$\text{II.}$ $\qquad f(x) = \left\{ \begin{array}{lr} 1,\qquad\qquad~~~ 1 \leq x <2\\ x^2-4x+5,~~~~2 \leq x \leq 3 \end{array} \right.$ $~~~~~\text{on}~~~~[1,3]$
C
$\text{III.}$ $~~~~~ \displaystyle f(x) = \big|x^2-1\big| ~~~~ \text{on} ~~~~[0,2]$

The graphs are shown to aid with your inquiry.

Skill check: Mean value theorem, L'Hopital, and local linearization