Question

The function y = a log x+bx² + x has extreme values at x=1 and x=2. Find a and b

Answer 1

$$\begin{gathered} \mathrm{Given}:f\left(x\right)=y=a\log x+b{x}^2+x \\ \Rightarrow f\text{'}\left(x\right)=\frac{a}{x}+2bx+1 \\ \mathrm{Since},f\text{'}\left(x\right)\mathrm{has}\mathrm{extreme}\mathrm{value}s\mathrm{at}x=1\mathrm{and}x=2,f\text{'}\left(1\right)=0. \\ \Rightarrow \frac{a}{1}+2b\left(1\right)+1=0 \\ \Rightarrow a=-1-2b...\left(1\right) \\ f\text{'}\left(2\right)=0 \\ \Rightarrow \frac{a}{2}+2b\left(2\right)+1=0 \\ \Rightarrow a+8b=-2 \\ \Rightarrow a=-2-8b...\left(2\right) \\ \mathrm{From}\mathrm{eqs}.\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get} \\ -2-8b=-1-2b \\ \Rightarrow 6b=-1 \\ \Rightarrow b=\frac{-1}{6} \\ \mathrm{Substituting}b=\frac{-1}{6}\mathrm{in}\mathrm{eq}.\left(1\right),\mathrm{we}\mathrm{get} \\ a=-1+\frac{1}{3}=\frac{-2}{3} \end{gathered}$$