#### Explain Solution for RD Sharma Maths Class 12 Chapter 17 Maxima and Minima Exercise Fill in the blanks Question 10 maths textbook solution

$(-2 / 3,-1 / 6)$

Hint:

For maxima or minima we must have ${f}'\left ( x \right )=0$

Given:

$y=a\log x+bx^{2}+x$  has its extreme values at $x=1$ & $x=2$

Solution:

\begin{aligned} &y=a \log x+b x^{2}+x \\ &y^{\prime}=\frac{a}{x}+2 b x+1 \\ &x=1 \\ &y^{\prime}=a+2 b+1=0 \\ &a+2 b=-1 \\ &a=-1-2 b \\ &x=2 \end{aligned}                                                                                                                           (1)

\begin{aligned} &y^{2}=\frac{a}{2}+4 b+1=0 \\ &a+8 b+2=0 \\ &a+8 b=-2 \\ &-1-2 b+8 b=-2 \\ &6 b=-2+1 \\ &6 b=-1 \\ &b=\frac{-1}{6} \end{aligned}                                                                                                                                    (2)

Substitute (2) in (1)

\begin{aligned} &a=-1-2\left(\frac{-1}{6}\right) \\ &a=\frac{-1-1}{3}=\frac{-2}{3} \\ &a=\frac{-2}{3}, b=\frac{-1}{6} \end{aligned}