Based on the rules provided in the photo, here's how we can analyze a function using critical points, the first derivative test, and the second derivative test.

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Example Problem:

Analyze $f(x) = x^3 - 3x^2 + 2$ to determine:

1. The critical points of $f(x)$,
2. Whether these points are local maxima, minima, or neither,
3. Any potential inflection points.

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Step 1: Find the critical points ($f'(x) = 0$)

The function is $f(x) = x^3 - 3x^2 + 2$. Compute the first derivative: $f'(x) = 3x^2 - 6x$

Factorize $f'(x)$: $f'(x) = 3x(x - 2)$

Set $f'(x) = 0$ to find the critical points: $3x(x - 2) = 0 \quad \implies \quad x = 0 \quad \text{or} \quad x = 2$

The critical points are at $x = 0$ and $x = 2$.

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Step 2: Use the second derivative test to classify critical points

Compute the second derivative: $f''(x) = 6x - 6$

Evaluate $f''(x)$ at each critical point:

• For $x = 0$: $f''(0) = 6(0) - 6 = -6 \quad (\text{negative, so a local maximum at } x = 0).$

• For $x = 2$: $f''(2) = 6(2) - 6 = 6 \quad (\text{positive, so a local minimum at } x = 2).$

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Step 3: Identify inflection points

To find potential inflection points, solve $f''(x) = 0$: $6x - 6 = 0 \quad \implies \quad x = 1$

Check the sign change of $f''(x)$ around $x = 1$:

• For $x < 1$ (e.g., $x = 0$): $f''(0) = -6$ (negative).
• For $x > 1$ (e.g., $x = 2$): $f''(2) = 6$ (positive).

Since $f''(x)$ changes sign, $x = 1$ is an inflection point.

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Summary of Results:

1. Critical points:

   • $x = 0$: Local maximum ($f(0) = 2$).
   • $x = 2$: Local minimum ($f(2) = -2$).

2. Inflection point:

   • $x = 1$: Inflection point ($f(1) = 0$).

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Would you like to attempt a similar question or see this visualized on a graph?

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Related Questions:

1. How would the critical points change for $f(x) = x^4 - 4x^3 + 6x^2 - 1$?
2. Can you classify the nature of $x = 0$ and $x = 2$ without using the second derivative test?
3. What are the implications of $f''(x) = 0$ for concavity and inflection points?
4. How does the first derivative test compare to the second derivative test in this case?
5. Can you compute the derivative of $g(x) = x^3 - 3x^2 + \ln(x)$ and analyze it?

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Tip:

When applying the second derivative test, remember that if $f''(x) = 0$, the test is inconclusive, and the first derivative test may be needed for clarification.