Question

Let f(x) = $\left\{ \begin{array} { c c } 4 x - x ^ { 3 } + \ln \left( a ^ { 2 } - 3 a + 3 \right) , & 0 \leq x < 3 \\ x - 18 , & x \geq 3 \end{array} \right.$

Complete set of 'a such that f(x) has a local minima at x = 3, is

• A. [-1, 2]
• B. (-∞, 1) ∪ (2, ∞)
• C. [1, 2]
• D. ) (-∞, -1) ∪ (2, ∞)

Answer 1

Answer: (C)

$1, 2$

Answer 2

Clearly f(x) is decreasing just before x = 3 and increasing after x = 3. For x = 3 to be the point of local minima.

f(3) ≥ f(3 – 0)

⇒ −15 ≥ 12 – 27+ ln (1² - 3a + 3) ⇒ 0 < a² – 3a + 3 < 1

⇒ 1 < a < 2