Question

The maxima and minima of the function $2x^3 - 15x^2 + 36x + 10$ occur respectively at

• A. $x=1, x=3$
• B. $x=2, x=1$
• C. $x=3, x=2$
• D. $x=1, x=2$
• E. $x=2, x=3$

Hint:

Second derivative test quickly identifies maxima/minima.

Answer 1

The Correct Option is C

Step 1: First derivative \[ \frac{dy}{dx} = 6x^2 - 30x + 36 \]

Step 2: Critical points \[ 6(x^2 -5x +6)=0 \Rightarrow (x-2)(x-3)=0 \] \[ x=2,3 \]

Step 3: Second derivative \[ \frac{d^2y}{dx^2} = 12x -30 \]

Step 4: Nature of points At $x=2$: \[ 12(2)-30 = -6 <0 \Rightarrow \text{maximum} \] At $x=3$: \[ 12(3)-30 = 6 >0 \Rightarrow \text{minimum} \]

Step 5: Arrange (max, min) \[ \text{Maximum at } x=2, \text{Minimum at } x=3 \] \[ \boxed{x=2, x=3} \]