Let g(x)


Let $\mathrm{g}(\mathrm{x})=\int_{0}^{x} f(\mathrm{t}) \mathrm{dt}$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3} \leq f(t) \leq 1$ for all $t \in[0,1]$ and $0 \leq f(t) \leq \frac{1}{2}$ for all $t \in(1,3]$ The largest possible interval in which $\mathrm{g}(3)$ lies is:

  1. (1) $\left[-1,-\frac{1}{2}\right]$

  2. (2) $\left[-\frac{3}{2},-1\right]$

  3. (3) $\left[\frac{1}{3}, 2\right]$

  4. (4) $[1,3]$

Correct Option: , 3


$\frac{1}{3} \leq f(\mathrm{t}) \leq 1 \forall \mathrm{t} \in[0,1]$

$0 \leq f(\mathrm{t}) \leq \frac{1}{2} \forall \mathrm{t} \in(1,3]$

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