Given f(x) = _ - 2 ^x t.g'(t) ,dt for x -2, where g is an increas… - Vidyadip

MCQ

Given $f(x) = \int\limits_{ - 2}^x {t.g'(t)\,dt} $ for $x \geq -2$, where $g$ is an increasing function, then

A
$ƒ(x)$ is an increasing function of $x$.
B
$ƒ(x)$ is a decreasing function of $x.$
✓
$ƒ(x)$ is increasing for $x > 0$ and decreasing for $x \in [-2,0)$.
D
None of these

✓

Answer

Correct option: C.

$ƒ(x)$ is increasing for $x > 0$ and decreasing for $x \in [-2,0)$.

c
$f^{\prime}(x)=x \cdot g^{\prime}(x)$

$\because g^{\prime}(x) \geq 0 \quad \forall x \geq-2$

$\therefore $ $f(\mathrm{x}) \downarrow$ for $[-2,0)$ and $\uparrow$ for $\mathrm{x}>0$

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