Second Derivative Test

Concavity and Critical Point Classification

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The second derivative test is a calculus tool used to classify critical points of a function as local maxima, local minima, or inconclusive cases. It matters because many optimization problems depend on knowing whether a point gives the highest value nearby, the lowest value nearby, or neither. This test connects algebraic derivatives to the visual shape of a graph. It gives a fast way to interpret how a curve bends near a critical point.

To use the test, first find critical points where $f'(x) = 0$ or where $f'(x)$ is undefined. Then evaluate the second derivative at those points. If $f''(c) > 0$ , the graph is concave up near $x = c$ , so $f(c)$ is a local minimum. If $f''(c) < 0$ , the graph is concave down near $x = c$ , so $f(c)$ is a local maximum, and if $f''(c) = 0$ , the test does not decide and more analysis is needed.

Key Facts

A critical point occurs where $f'(c) = 0$ or $f'(c)$ is undefined, provided $c$ is in the domain of $f$ .
Second derivative test: if $f'(c) = 0$ and $f''(c) > 0$ , then $f(c)$ is a local minimum.
Second derivative test: if $f'(c) = 0$ and $f''(c) < 0$ , then $f(c)$ is a local maximum.
If $f'(c) = 0$ and $f''(c) = 0$ , the second derivative test is inconclusive.
Concavity rule: $f''(x) > 0$ means the graph is concave up, and $f''(x) < 0$ means the graph is concave down.
At a local maximum the tangent is horizontal and the curve bends downward nearby, while at a local minimum the tangent is horizontal and the curve bends upward nearby.

Vocabulary

Critical point: A point on the graph where the first derivative is zero or undefined and the function is defined there.
Second derivative: The derivative of the first derivative, which describes how the slope is changing.
Concavity: The direction a graph bends, either upward or downward, over an interval.
Local maximum: A point where the function value is greater than nearby function values.
Local minimum: A point where the function value is less than nearby function values.

Common Mistakes to Avoid

Using the second derivative test at a point that is not critical, which is wrong because the test only applies after verifying $f'(c) = 0$ .
Concluding that $f''(c) = 0$ means there is no maximum or minimum, which is wrong because the test is only inconclusive there and the point could still be a max, min, or neither.
Mixing up the sign of $f''(c)$ , which is wrong because $f''(c) > 0$ indicates concave up and a local minimum, while $f''(c) < 0$ indicates concave down and a local maximum.
Forgetting to check that the function is defined at the point, which is wrong because an undefined point cannot be classified as a local maximum or minimum.

Practice Questions

1 For $f(x) = x^2 - 4x + 1$ , find the critical point and use the second derivative test to classify it.
2 For $f(x) = x^3 - 3x^2 + 2$ , find all critical points and use the second derivative test on each one.
3 A function has $f'(2) = 0$ and $f''(2) = 0$ . Explain why the second derivative test does not classify $x = 2$ and describe what other information you could check next.

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