First slide

Mean value theorems

Question

$Given f (x) = 4 - {(\frac{1}{2} - x)}^{2 / 3}, g (x) = \{\begin{array}{ll} \frac{\tan [x]}{x}, x \neq 0 \\ 1, x = 0 \end{array},$ $h (x) = {x}, k (x) = 5^{\log_{2} (x + 3)}$
then in [0, 1] Lagrange's mean value theorem is NOT applicable to the (where [.] and { .} represents greatest integer
functions and fractional part functions, respectively)

Difficult

A

f
B

g
C

k
D

h

Solution

$f is not differentiable at x = \frac{1}{2}$
$g is not continuous in [0, 1] at x = 0$
$h is not continuous in [0, 1] at x = 1$
$k (x) = (x + 3)^{\ln_{2} 5} = (x + 3) p, where 2 < p < 3, which is continuous and differentiable$ .

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