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Theorems for Differentiability

Given fx= 4...

Question

Given $f (x) = 4 - (\frac{1}{2} - x)^{\frac{2}{3}}$ , $g (x) = {\begin{matrix} \frac{t a n [x]}{x}, x \neq 0 1, x = 0 \end{matrix}, h (x) = x, k (x) = 5^{{log}_{2} (x + 3)}$
Then in $[0, 1]$ , Lagrange's mean value theorm is not applicable to (where [.] and {.} represents the greatest integer functions and fractional part functions, respectively)

f

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g

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k

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h

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Solution

The correct options are
A $f$
B $g$
For g(x) langrange theorem do not apply because integer function is not continous and differentiable.
Also for f(x) the derivative f'(x) is cont continuous.
all other function are differentiable and continous.
Answer is A, B only.

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