Question

Find the value(s) of a for which f(x) = x³ − ax is an increasing function on R.

Answer 1

$$\begin{gathered} f\left(x\right)=x^3-ax \\ f\text{'}\left(x\right)=3x^2-a \\ \text{Given:}f\left(x\right)\text{is increasing on}\mathit{\text{R.}} \\ \Rightarrow f\text{'}\left(x\right)\ge 0\forall x\in R \\ \Rightarrow 3x^2-a\ge 0\forall x\in R \\ \Rightarrow a\le 3x^2\forall x\in R \\ \text{The least value of 3}{\mathit{\text{x}}}^2\text{is 0.} \\ \text{∴}a\le 0 \end{gathered}$$