Question:
A point of inflection of the curve given by $y = x^3 - 6x^2 + 12x + 50$ occurs when
A point of inflection of the curve given by $y = x^3 - 6x^2 + 12x + 50$ occurs when
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Always verify sign change to confirm inflection point.
Updated On: Apr 30, 2026
- $x=2$
- $x=\frac{3}{2}$
- $x=2$
- $x=3$
- $x=0$
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The Correct Option is A
Solution and Explanation
Concept:
Point of inflection occurs where:
\[
\frac{d^2y}{dx^2} = 0 \text{and sign changes}
\]
Step 1: First derivative. \[ \frac{dy}{dx} = 3x^2 - 12x + 12 \]
Step 2: Second derivative. \[ \frac{d^2y}{dx^2} = 6x - 12 \]
Step 3: Set equal to zero. \[ 6x - 12 = 0 \Rightarrow x = 2 \]
Step 4: Check sign change. For $x<2$: negative
For $x>2$: positive Thus inflection point exists. \[ \boxed{x=2} \]
Step 1: First derivative. \[ \frac{dy}{dx} = 3x^2 - 12x + 12 \]
Step 2: Second derivative. \[ \frac{d^2y}{dx^2} = 6x - 12 \]
Step 3: Set equal to zero. \[ 6x - 12 = 0 \Rightarrow x = 2 \]
Step 4: Check sign change. For $x<2$: negative
For $x>2$: positive Thus inflection point exists. \[ \boxed{x=2} \]
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