First slide

Maxima and minima

Question

$If f (x) = a \log_{e} | x | + b x^{2} + x has extremums at x = 1 and x = 3, then$

NA

A

$a = \frac{- 3}{4}, b = \frac{- 1}{8}$
B

$a = \frac{3}{4}, b = \frac{- 1}{8}$
C

$a = \frac{- 3}{4}, b = \frac{1}{8}$
D

$a = \frac{1}{2}, b = - 1$

Solution

$\begin{array}{l} at x = 1, 3 we have | x | = x \\ ∴ f (x) = a \log_{e} x + b x^{2} + x \\ ∴ f^{'} (x) = \frac{a}{x} + 2 b x + 1 \end{array}$
$\begin{array}{l} According to question, f^{'} (1) = 0, f^{'} (3) = 0 \\ ∴ a + 2 b + 1 = 0, \frac{a}{3} + 6 b + 1 = 0 \end{array}$
$On solving, we get a = \frac{- 3}{4}, b = - \frac{1}{8}$

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