SATHEE: Application of Derivatives 4 Question 16

Application of Derivatives 4 Question 16

17. If $x = - 1$ and $x = 2$ are extreme points of $f (x) = α \log | x | + β x^{2} + x$ , then

(a) $α = - 6, β = \frac{1}{2}$

(b) $α = - 6, β = - \frac{1}{2}$

(d) $α = 2, β = \frac{1}{2}$

(2014 Main)

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Answer:

Correct Answer: 17. (c)

Solution:

Here, $x = - 1$ and $x = 2$ are extreme points of $f (x) = α \log | x | + β x^{2} + x$ , then

$\begin{aligned} f^{'} (x) & = \frac{α}{x} + 2 β x + 1 \\ f^{'} (- 1) & = - α - 2 β + 1 = 0 \end{aligned}$

[at extreme point, $f^{'} (x) = 0$ ]

$f^{'} (2) = \frac{α}{2} + 4 β + 1 = 0$

On solving Eqs. (i) and (ii), we get

$α = 2, β = - \frac{1}{2}$

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Application of Derivatives 4 Question 16

17. If x=−1 and x=2 are extreme points of f(x)=αlog⁡|x|+βx2+x, then

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17. If $x = - 1$ and $x = 2$ are extreme points of $f (x) = α \log | x | + β x^{2} + x$ , then