Question

If the function f(x) = 2x² − kx + 5 is increasing on [1, 2], then k lies in the interval
(a) (−∞, 4)
(b) (4, ∞)
(c) (−∞, 8)
(d) (8, ∞)

Answer 1

(a) (−∞, 4)

$$\begin{gathered} f\left(x\right)=2x^2-kx+5 \\ f\text{'}\left(x\right)=4x-k \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow 4x-k>0 \\ \Rightarrow k<4x \\ \text{Since}x\in \left[1,2\right],4x\in \left[4,8\right]. \\ \text{So, the minimum value of 4}\mathit{\text{x}}\text{is 4.} \\ \mathrm{Since}k<4x,k<4. \\ \Rightarrow k\in \left(-\infty ,4\right) \end{gathered}$$