Finding Local Maxima and Minima Using Derivative Tests

When studying a function in calculus, it's important to find points where the function is at a local high (maximum) or local low (minimum). We can do this using something called the first and second derivative tests. These tests help us understand how the function behaves at different points.

What Are Critical Points?

1. Critical Points Defined:

   • A critical point happens at $x = c$ when the derivative of the function $f'(x)$ is zero (meaning it stops changing) or when the derivative doesn’t exist. These points might be where the function has a maximum or minimum.

2. Finding Derivatives:

   • To find critical points, you first need to calculate the derivative of the function. If we have a function $f(x)$ that is smooth (meaning it can be drawn without lifting your pencil from the paper), you find $f'(x)$.

3. Setting the Derivative to Zero:

   • Now, set $f'(x) = 0$ to find the possible locations of local extrema. Also, look at points where $f'(x)$ doesn't exist.

4. Looking Within the Function's Domain:

   • It's important to only check within the function's domain (the values of $x$ the function can take), and remember any points where the function might break or change.

First Derivative Test

1. Using the First Derivative Test:

   • The first derivative test helps us figure out if a critical point is a local maximum, local minimum, or neither. Here's how it works:

2. Select Intervals:

   • After finding critical points, choose test points in the intervals around those points. If our critical point is $c$, we look at the intervals $(-\infty, c)$ and $(c, +\infty)$.

3. Check the Sign of the Derivative:

   • Evaluate $f'(x)$ at these test points:
     • If $f'(x) > 0$ (positive) before $c$ and $f'(x) < 0$ (negative) after $c$, then $f(c)$ is a local maximum.
     • If $f'(x) < 0$ before $c$ and $f'(x) > 0$ after $c$, then $f(c)$ is a local minimum.
     • If $f'(x)$ stays the same on both sides of $c$, then $f(c)$ is neither a maximum nor a minimum.

Example of First Derivative Test

Let’s look at the function $f(x) = x^3 - 3x^2 + 4$. First, we find the derivative:

$$f'(x) = 3x^2 - 6x.$$

Next, set the derivative equal to zero:

$$3x^2 - 6x = 0 \implies 3x(x - 2) = 0 \implies x = 0 \text{ or } x = 2.$$

Now we test the intervals around these critical points.

• For $x < 0$: Use $x = -1$, then $f'(-1) = 9 > 0$ (positive).
• For $0 < x < 2$: Use $x = 1$, then $f'(1) = -3 < 0$ (negative).
• For $x > 2$: Use $x = 3$, then $f'(3) = 9 > 0$ (positive).

From these evaluations:

• At $x = 0$, $f$ goes from positive to negative, showing it is a local maximum.
• At $x = 2$, $f$ goes from negative to positive, showing it is a local minimum.

Second Derivative Test

1. Using the Second Derivative Test:

   • The second derivative test is another way to classify critical points. We find the second derivative $f''(x)$:
   • The test is straightforward:
     • If $f''(c) > 0$, then $f(c)$ is a local minimum.
     • If $f''(c) < 0$, then $f(c)$ is a local maximum.
     • If $f''(c) = 0$, we can’t decide, and we may need to use the first derivative test.

2. Example Calculation of Second Derivative: For the earlier function $f(x) = x^3 - 3x^2 + 4$, we find the second derivative:

$$f''(x) = 6x - 6.$$

• Checking the second derivative at our critical points:
  • For $x = 0$: $f''(0) = -6 < 0 \implies \text{local maximum at } x = 0.$
  • For $x = 2$: $f''(2) = 6 > 0 \implies \text{local minimum at } x = 2.$

Summary

In conclusion, finding local maxima and minima using the first and second derivative tests involves:

• Finding critical points by setting the first derivative to zero and studying how the derivative behaves around these points.
• Using the first derivative test to see if the signs change around the critical points, which helps us identify whether they are local maxima, minima, or neither.
• Optionally, using the second derivative test for a more straightforward classification based on concavity.

These tests are essential tools in calculus for understanding how functions behave, helping in graphing, solving problems, and learning advanced math concepts. By mastering these methods, students can gain valuable insights into how different functions work.