The First Derivative Test is a method for deciding whether a critical point is a local maximum, local minimum, or neither. It uses the sign of the derivative on intervals around a critical point instead of relying only on the graph shape. This matters because it connects slope behavior to function behavior in a precise and testable way.
It is one of the main tools students use to analyze graphs in calculus.
A critical point occurs where or where does not exist, as long as is in the domain of . To apply the test, find intervals on each side of the critical point and determine whether is positive or negative there. If changes from positive to negative, the function changes from increasing to decreasing, so there is a local maximum.
If changes from negative to positive, the function changes from decreasing to increasing, so there is a local minimum.
Understanding First Derivative Test
A sign chart is a compact record of what the derivative does across the number line. Start by finding every value that can split the domain into separate intervals. These values include derivative zeros, places where the derivative is undefined, and any points missing from the original function.
Put them in numerical order. Then choose one simple test value from each open interval and evaluate the derivative there. The exact derivative value usually does not matter.
Only its sign matters. This saves work because a continuous derivative cannot switch from positive to negative inside an interval without passing through zero or becoming undefined.
Factoring the derivative often makes the sign chart much easier to build. Consider a derivative made from factors such as x minus two, x plus one, and x minus three squared. Each linear factor changes sign when its zero is crossed.
A squared factor does not change sign because its value stays nonnegative on both sides. This pattern explains why some stationary points are turning points while others are not. An odd power usually causes a sign change.
An even power usually does not. Students should still test intervals rather than relying only on this shortcut, especially when fractions, roots, or complicated factors appear.
A derivative can be undefined for important reasons. A denominator can become zero, or a function can have a sharp corner. First check whether the original function exists at that input.
If it does not exist, there cannot be a local maximum or minimum there because there is no function value to classify. If the function exists but its derivative fails to exist, the point may still be an extremum. The absolute value function has a sharp lowest point at zero even though its derivative is undefined there.
Domain restrictions matter just as much. For a function defined only on a limited interval, an endpoint needs separate attention because it has values on only one side.
The test describes local behavior, not the highest or lowest value across an entire domain. A local maximum is only higher than nearby function values. To find an absolute maximum or minimum on a closed interval, compare the function values at all critical points inside the interval and at both endpoints.
This is useful in optimization problems. A business model may use a derivative to locate the production level with the greatest profit. A physics model may use it to find when position reaches a highest point or when speed stops increasing.
Keep the number line, derivative signs, and function behavior clearly separated. The derivative sign belongs in the sign chart. The conclusion describes the original function.
Key Facts
- Critical points occur where or does not exist, with in the domain of .
- If on an interval, then is increasing on that interval.
- If on an interval, then is decreasing on that interval.
- If changes from to at , then is a local maximum.
- If changes from to at , then is a local minimum.
- If does not change sign at , then is not a local extremum by the First Derivative Test.
Vocabulary
- Derivative
- The derivative measures the instantaneous rate of change or slope of the function at .
- Critical point
- A critical point is a value in the domain where or does not exist.
- Local maximum
- A local maximum is a point where the function value is greater than nearby function values.
- Local minimum
- A local minimum is a point where the function value is less than nearby function values.
- Sign chart
- A sign chart is a diagram that shows whether is positive or negative on intervals.
Common Mistakes to Avoid
- Assuming every point where is a max or min, because some critical points have no sign change and are neither. Always test the sign of on both sides.
- Forgetting that points where does not exist can still be critical points, because the derivative being undefined does not remove the point from consideration.
- Using the function value instead of the derivative sign, because the First Derivative Test depends on whether slopes are positive or negative. Build a sign chart for , not for .
- Testing only one side of a critical point, because a local extremum depends on how the derivative behaves before and after the point. You need intervals on both sides to classify it correctly.
Practice Questions
- 1 Let . Find the critical points and use the First Derivative Test to classify each one.
- 2 A function has derivative . Find all critical points and determine where the function is increasing, decreasing, and whether each critical point is a local extremum.
- 3 A function has a critical point at , and is negative on both sides of . What does the First Derivative Test say about , and why?