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Question:

Let a, b ∈ R be such that the function f given by f(x) = ln|x| + bx² + ax, x ≠ 0 has extreme values at x = -1 and x = 2. Statement 1: f has local maximum at x = -1 and at x = 2. Statement 2: a = 1/2, b = -1/4. Select the correct option regarding the truth values of Statement 1 and Statement 2.

Statement 1 is false, Statement 2 is true

Statement 1 is true, Statement 2 is true; Statement 2 is correct explanation for Statement 1

Statement 1 is true, Statement 2 is false.

Statement 1 is true, Statement 2 is true; Statement 2 is not a correct expalnation for Statement 1.

Solution:

f′(x) = 1/x + 2bx + a
f has extreme values and is differentiable. ⇒ f′(-1) = 0 ⇒ -a - 2b = 1
⇒ f′(2) = 0 ⇒ a + 4b = -1/2
⇒ a = 1/2, b = -1/4
f′′(-1) and f′′(2) are negative. Therefore, f has local maxima at -1 and 2. Hence, option 'B' is correct.