Mark the correct alternative in the following:
Question:

Mark the correct alternative in the following:

If the function $f(x)=2 x^{2}-k x+5$ is increasing on $[1,2]$, then $k$ lies in the interval.

A. $(-\infty, 4)$

B. $(4, \infty)$

C. $(-\infty, 8)$

D. $(8, \infty)$

Solution:

Formula:- The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$

$f(x)=2 x^{2}-k x+5$

$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=4 \mathrm{x}-\mathrm{k}=\mathrm{f}^{\prime}(\mathrm{x})$

$f^{\prime}(x)>0$

$\Rightarrow 4 x-k>0$

$\Rightarrow K<4 x$

For $x=1$

$\Rightarrow K<4$