Question

If the function $f\left(x\right)=2x^2-kx+5$ is increasing on $[1,2]$, then $k$ lies in the interval

• A. $(-\infty ,4)$
• B. $(4,\infty )$
• C. $(-\infty ,8)$
• D. $(8,\infty )$

Answer 1

The correct option is A $(-\infty ,4)$

$f\left(x\right)=2x^2-kx+5$

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

For $f\left(x\right)$ to be increasing we must have

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

$\Rightarrow 4x-k>0$

$k<4x$

since $x\in [1,2]\Rightarrow 4x\in [4,8]$

So the minimum value of $4x$ is $4$

Since $k<4x\Rightarrow k<4$

$\Rightarrow k\in (-\infty ,4)$.