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Question:
Mark the correct alternative in the following:
If the function $f(x)=x^{2}-k x+5$ is increasing on $[2,4]$, then
A. $k \in(2, \infty)$
B. $k \in(-\infty, 2)$
C. $k \in(4, \infty)$
D. $k \in(-\infty, 4)$
Solution:
Formula:- (i) 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 $x \in(a, b)$
$f(x)=x^{2}-k x+5$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=2 \mathrm{x}-\mathrm{k}=\mathrm{f}(\mathrm{x})$
For increasing function $f^{\prime}(x)>0$
$2 x-k>0$
$\Rightarrow K<2 x$
Putting $x=2$
$K<4$
$\Rightarrow K \in(-\infty, 4)$
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