To determine whether a critical point (where the derivative equals zero) is a minimum, maximum, or neither, you use the Second Derivative Test or the First Derivative Test:

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1. Second Derivative Test:

1. Find the second derivative of the function $f''(x)$.
2. Evaluate the second derivative at the critical point $c$, where $f'(c) = 0$:
   • If $f''(c) > 0$, the point is a local minimum (concave up at $c$).
   • If $f''(c) < 0$, the point is a local maximum (concave down at $c$).
   • If $f''(c) = 0$, the test is inconclusive. In this case, use another method (e.g., First Derivative Test or analyze the behavior of $f'(x)$).

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2. First Derivative Test:

1. Examine the sign of the first derivative $f'(x)$ on either side of the critical point $c$:
   • If $f'(x)$ changes from positive to negative as $x$ passes through $c$, the point is a local maximum.
   • If $f'(x)$ changes from negative to positive as $x$ passes through $c$, the point is a local minimum.
   • If $f'(x)$ does not change sign, $c$ is neither a maximum nor a minimum (it could be an inflection point).

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Examples:

1. Second Derivative Test Example:

   • $f(x) = x^3 - 3x^2 + 1$
   • $f'(x) = 3x^2 - 6x = 3x(x - 2)$ → critical points: $x = 0$ and $x = 2$
   • $f''(x) = 6x - 6$
     • At $x = 0$: $f''(0) = 6(0) - 6 = -6$ → local maximum.
     • At $x = 2$: $f''(2) = 6(2) - 6 = 6$ → local minimum.

2. First Derivative Test Example:

   • Same $f'(x) = 3x(x - 2)$.
     • For $x = 0$: $f'(x)$ changes from positive (left of 0) to negative (right of 0) → local maximum.
     • For $x = 2$: $f'(x)$ changes from negative (left of 2) to positive (right of 2) → local minimum.

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Would you like to see further examples or clarifications? Here are related questions to explore:

1. What if the second derivative is zero at a critical point?
2. How do you use concavity to determine inflection points?
3. Can there be absolute minima/maxima in addition to local ones?
4. How do you handle functions where derivatives are not continuous?
5. How does this relate to real-world optimization problems?

Tip: Always plot the function if possible; visualizing behavior can confirm analytical results.