To determine local minima and local maxima, I follow a mathematical procedure rooted in calculus. Here's the step-by-step breakdown:

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1. Find the First Derivative

• The first derivative of a function $f(x)$, denoted as $f'(x)$, represents the slope of the function at each point.
• To find critical points, I solve $f'(x) = 0$, because at these points, the slope of the tangent to the curve is zero (i.e., the function levels off).

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2. Identify Critical Points

• The solutions to $f'(x) = 0$ are potential candidates for local maxima, local minima, or inflection points.
• I also consider points where $f'(x)$ is undefined, as these could also indicate critical points.

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3. Use the Second Derivative Test

• To classify the critical points as local maxima or minima, I calculate the second derivative $f''(x)$, which measures the concavity of the function:
  • If $f''(x) > 0$ at a critical point, the function is concave up, and the point is a local minimum.
  • If $f''(x) < 0$ at a critical point, the function is concave down, and the point is a local maximum.
  • If $f''(x) = 0$, the test is inconclusive, and I may use other methods like the First Derivative Test.

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4. First Derivative Test (if necessary)

• I examine the sign of $f'(x)$ on intervals around the critical points:
  • If $f'(x)$ changes from positive to negative at a critical point, it’s a local maximum.
  • If $f'(x)$ changes from negative to positive at a critical point, it’s a local minimum.

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5. Boundary Points (if applicable)

• If the function is defined on a closed interval, I evaluate $f(x)$ at the endpoints of the interval as well, since local extrema can occur there.

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Example:

For the function $f(x) = x^3 - 3x^2 + 4$:

1. First derivative: $f'(x) = 3x^2 - 6x$
2. Solve $f'(x) = 0$: $3x(x - 2) = 0$, so $x = 0$ and $x = 2$.
3. Second derivative: $f''(x) = 6x - 6$
   • At $x = 0$: $f''(0) = -6$, so $x = 0$ is a local maximum.
   • At $x = 2$: $f''(2) = 6$, so $x = 2$ is a local minimum.

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Would you like further details, a worked-out example, or clarification on this method?

Follow-Up Questions:

1. Can you show an example of finding local extrema for a specific function?
2. What happens if $f''(x) = 0$ at a critical point?
3. How do you find extrema for multivariable functions (e.g., $f(x, y)$)?
4. What is the significance of inflection points in this context?
5. How does this process differ for non-polynomial functions, like trigonometric or exponential ones?

Tip:

Always check both the critical points and endpoints (if defined on a closed interval) when finding extrema!