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Maxima and Minima — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMaxima and Minima4 Marks
Question
The function y = a log x + bx2 + x has extreme values at x = 1 and x = 2. Find a and b.
✓

Answer

We have, $\text{y}=\text{a}\log\text{x}+\text{b}\text{x}^{2}=\text{x}$ 
$\frac{\text{dy}}{\text{dx}}=\frac{\text{a}}{\text{x}}+2\text{b}\text{x}+1$
and $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\frac{-\text{a}}{\text{x}^{2}}+2\text{b}$
For maxima and minimum value,
$\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\frac{\text{a}}{\text{x}}+2\text{bx}+1=0$
Given that extreme value exist at x = 1, 2
$\Rightarrow\text{a}+2\text{b}=-1\ ...(\text{i})$
$\frac{\text{a}}{2}+4\text{b}=-1$
$\Rightarrow\text{a}+8\text{b}=-2\ ...(\text{ii})$
Solving (i) and (ii), We get
$\text{a}=\frac{-2}{3}, \ \text{b}=\frac{-1}{6}$

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