Question

Find the least value of a such that the function f given is strictly increasing on [1, 2].

Answer 1

We have,

Now, function f is increasing on [1,2].

$$\begin{gathered} \therefore f\text{'}\left(x\right)\ge 0\mathrm{on}\left[1,2\right] \\ \mathrm{Now},\mathrm{we}\mathrm{have}1⩽x⩽2 \\ \Rightarrow 2⩽2x⩽4 \\ \Rightarrow 2+a⩽2x+a⩽4+a \\ \Rightarrow 2+a⩽f\text{'}\left(x\right)⩽4+a \\ \mathrm{Since}f\text{'}\left(x\right)\ge 0 \\ \Rightarrow 2+a\ge 0 \\ \Rightarrow a\ge -2 \\ \mathrm{So},\mathrm{least}\mathrm{value}\mathrm{of}a\mathrm{is}-2. \end{gathered}$$