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JEE Main 2014

JEE Main 2014 Mathematics question on Application of Derivatives

If x=−1x = -1 and x=2x = 2 are extreme points of f(x)=αlog⁡∣x∣+βx2+xf(x) = \alpha \log|x| + \beta x^2 + x, then
  1. A.
    α=−6, β=12\alpha = -6,\ \beta = \dfrac{1}{2}
  2. B.
    α=−6, β=−12\alpha = -6,\ \beta = -\dfrac{1}{2}
  3. C.
    α=2, β=−12\alpha = 2,\ \beta = -\dfrac{1}{2}
  4. D.
    α=2, β=12\alpha = 2,\ \beta = \dfrac{1}{2}
Show answer and solution

Correct option: C

Solution

At extreme points, f'(x) = 0. We have f'(x) = α/x + 2βx + 1.

At x = -1: -α - 2β + 1 = 0 → α + 2β = 1
At x = 2: α/2 + 4β + 1 = 0 → α + 8β = -2

Subtracting the first from the second: 6β = -3 → β = -1/2, then α = 1 - 2(-1/2) = 2.

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