Question

For the function $f(x) = x^3 - 6x^2 + 12x - 3$, $x = 2$ is:}

• A. A point of minimum
• B. A point of inflection
• C. Not a critical point
• D. A point of maximum

Answer 1

The Correct Option is B

1. Understand the problem:

We need to analyze the nature of the point \( x = 2 \) for the function \( f(x) = x^3 - 6x^2 + 12x - 3 \).

2. Find the first derivative:

\[ f'(x) = \frac{d}{dx}(x^3 - 6x^2 + 12x - 3) = 3x^2 - 12x + 12 \]

3. Identify critical points:

Set \( f'(x) = 0 \):

\[ 3x^2 - 12x + 12 = 0 \implies x^2 - 4x + 4 = 0 \implies (x-2)^2 = 0 \implies x = 2 \]

Thus, \( x = 2 \) is a critical point.

4. Find the second derivative:

\[ f''(x) = \frac{d}{dx}(3x^2 - 12x + 12) = 6x - 12 \]

5. Evaluate the second derivative at \( x = 2 \):

\[ f''(2) = 6(2) - 12 = 0 \]

Since \( f''(2) = 0 \), the second derivative test is inconclusive.

6. Analyze higher-order derivatives or behavior:

Compute the third derivative:

\[ f'''(x) = 6 \neq 0 \]

Since the lowest-order non-zero derivative at \( x = 2 \) is odd, \( x = 2 \) is a point of inflection.

Correct Answer: (B) A point of inflexion

Answer 2

Find $f'(x)$ and $f''(x)$: \[ f'(x) = 3x^2 - 12x + 12, \quad f''(x) = 6x - 12. \] At $x = 2$: \[ f'(2) = 0, \quad f''(2) = 0. \] Check $f'''(x)$: \[ f'''(x) = 6 \implies f'''(2) = 6 \neq 0. \] Thus, $x = 2$ is a point of inflection.