Let $g (x)$ is a real valued function which is continuous function on $[0, 2]$ and differentiable in $(0, 1)$ , then which of the following is / are always correct?

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$| g (b) | - | g (\frac{1}{10}) | \leq \int_{0}^{1} | g' (x) | d x \forall b \in (0, 1)$

$| g (b) | - | g (\frac{1}{10}) | \leq \int_{0}^{1} | g^{'} (x) | d x$ for atleast one value of b in $(0, 1)$

$\lim_{x \to 1} g (x)$ exist

$\lim_{x \to 1} [\frac{3}{2} g (x)]$ exist, when [A] denotes greatest integer less than or equal to A

answer is A, C, D.

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Detailed Solution

$g (x)$ continuous $\Rightarrow {lim}_{x \to 1} g (x)$ exist
Let $b \in (0, 1)$
$| \int_{1 / 10}^{b} (x) dx | \leq \int_{1 / 10}^{b} | (x) | dx \leq \int_{1 / 10}^{1} | g^{'} (x) | dx | g (b) - g (\frac{1}{10}) | \leq \int_{0}^{1} | g^{'} (x) | dx | g (b) | - | g (\frac{1}{10}) | \leq \int_{0}^{1} | g^{'} (x) | d x \forall b \in (0, 1) ∴ | g (b) | \leq \int_{0}^{1} | g^{'} (x) | dx + | g (\frac{1}{10}) | \leq \int_{0}^{1} | g^{'} (x) | dx + \int_{0}^{1} | g (x) | d x$