Question

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

Answer 1

(d) k ∈ (−∞, 4)

$$\begin{gathered} f\left(x\right)=x^2-kx+5 \\ f\text{'}\left(x\right)=2x-k \\ \mathrm{Given}:f\left(x\right)\mathrm{is}\mathrm{increasing}\mathrm{on}[2,4]. \\ \Rightarrow f\text{'}\left(x\right)>0 \\ \Rightarrow 2x-k>0 \\ \Rightarrow k<2x \\ \text{∵}x\in \left[2,4\right],\text{maximum value of}\mathit{\text{k}}\text{is 4,}\mathit{\text{k}}\text{< 4.} \\ \therefore k\in \left(-\infty ,4\right) \end{gathered}$$