Question

Set of value of b for which local extrema of the function f(x) are positive where f(x) = $\frac{2}{3}$a²x² – $\frac{5a}{2}$ x² + 3x + b and maxima occurs at x = 1/3 is

• A. (–4, ∞)
• B. $\left( - \frac{3}{8},\infty \right)$
• C. $\left( - 10,\frac{3}{8} \right)$
• D. None of these

Answer 1

Answer: (B)

$\left( - \frac{3}{8},\infty \right)$

Answer 2

f '(x) = 2a²x² – 5ax + 3 = (ax – 1) (2a x – 3) = 0

x = 1/a, 3/2a

If a > 0 then local maxima occurs at x = 1/a and minima at

x = 3/2a

Q maxima occurs x = 1/a = 1/3 ⇒ a = 3

minima occurs x = $\frac{3}{2a}$ = 1/2

f$\left( \frac{1}{2} \right)$ > 0

⇒ 3/8 + b > 0, b > – 3/8

If a < 0 then maxima shall occur at x = 3/2a and minima at

x = 1/2a $\frac{3}{2a} = \frac{1}{3}$ ⇒ a = $\frac{9}{2}$ > 0 not admissible

Hence b > – 3/8