Maxima/Minima Based Problems - Higher Order Derivative Test

When dealing with functions in calculus, finding the points at which a function reaches its maximum or minimum values is a common problem. These points are known as local maxima and minima. The higher order derivative test is a useful tool for determining whether a critical point (a point where the first derivative is zero or undefined) is a local maximum, local minimum, or neither.

Understanding Critical Points

Before we delve into the higher order derivative test, it's important to understand critical points. A critical point occurs where the first derivative of a function is zero or undefined. Mathematically, if $f'(x) = 0$ or $f'(x)$ does not exist, then $x$ is a critical point of $f(x)$.

First Derivative Test

The first derivative test can sometimes be used to determine the nature of a critical point. If the derivative changes from positive to negative at a critical point, the function has a local maximum there. If the derivative changes from negative to positive, the function has a local minimum. If the derivative does not change signs, the test is inconclusive.

Higher Order Derivative Test

When the first derivative test is inconclusive, or as an alternative method, the higher order derivative test can be used. This test involves taking the second derivative and possibly higher derivatives of the function at the critical points.

Second Derivative Test

The second derivative test states that:

• If $f''(x) > 0$ at a critical point, the function has a local minimum there.
• If $f''(x) < 0$ at a critical point, the function has a local maximum there.
• If $f''(x) = 0$, the test is inconclusive, and higher derivatives must be considered.

Higher Order Derivatives

When the second derivative test is inconclusive, we look at the third derivative, fourth derivative, and so on, until we find a non-zero derivative. The general rule is:

• If the first non-zero derivative at a critical point is of even order, then:
  • If the derivative is positive, the function has a local minimum.
  • If the derivative is negative, the function has a local maximum.
• If the first non-zero derivative at a critical point is of odd order, the function has an inflection point, and the test does not provide information about maxima or minima.

Table of Outcomes

Derivative Order	Derivative Sign	Outcome
2nd (even)	Positive	Local Minimum
2nd (even)	Negative	Local Maximum
2nd (even)	Zero	Inconclusive (check higher order)
3rd (odd)	Non-zero	Inflection Point
4th (even)	Positive	Local Minimum
4th (even)	Negative	Local Maximum

Examples

Let's go through some examples to illustrate the higher order derivative test.

Example 1: Second Derivative Test

Consider the function $f(x) = x^3 - 3x^2 + 2$.

1. Find the first derivative: $f'(x) = 3x^2 - 6x$.
2. Set the first derivative equal to zero to find critical points: $3x^2 - 6x = 0 \Rightarrow x = 0, 2$.
3. Find the second derivative: $f''(x) = 6x - 6$.
4. Evaluate the second derivative at the critical points:
   • $f''(0) = -6 < 0$, so there is a local maximum at $x = 0$.
   • $f''(2) = 6 > 0$, so there is a local minimum at $x = 2$.

Example 2: Higher Order Derivative Test

Consider the function $f(x) = x^4$.

1. Find the first derivative: $f'(x) = 4x^3$.
2. Set the first derivative equal to zero to find critical points: $4x^3 = 0 \Rightarrow x = 0$.
3. Find the second derivative: $f''(x) = 12x^2$.
4. Evaluate the second derivative at the critical point: $f''(0) = 0$, so the test is inconclusive.
5. Find the third derivative: $f'''(x) = 24x$.
6. Evaluate the third derivative at the critical point: $f'''(0) = 0$, still inconclusive.
7. Find the fourth derivative: $f''''(x) = 24$.
8. Evaluate the fourth derivative at the critical point: $f''''(0) = 24 > 0$, so there is a local minimum at $x = 0$.

Conclusion

The higher order derivative test is a powerful tool for analyzing the nature of critical points in a function. It provides a systematic way to determine local maxima and minima when the first derivative test is inconclusive. By understanding and applying this test, students can solve complex maxima/minima problems effectively in their exams.