113. Derivative Test

Find critical points by solving

Derivative Local Max Concave Down
Local Min Concave Up
Derivative Local Max Concave Down
Local Min Concave Up

113.0.1. First Derivative Test

If changes sign around a critical point , we can determine if is a local maximum or minimum:

Example

If changes from negative to positive at , then is a local minimum

Suppose:

Step 1. Find critical point

Solve :

So, is the only critical point

Step 2. Test sign of around

  • Left:
  • Right: :

Step 3. Interpretation

Since changes from negativetopositive at ,

Example

If changes from positive to negative at , then is a local maximum

Suppose:

Step 1. Find critical point

Solve

So, is the only critical point

Step 2. Test sign of around

  • Left:
  • Right:

Step 3. Interpretation

Since changes from positivetonegative at ,

Example

If does not change sign, is not a local extremum

Suppose

Step 1. Find critical point

Solve

So, is the only critical point

Step 2. Test sign of around

  • Left:
  • Right:

Step 3. Interpretation

Since does not change sign at ,

but rather a point of inflection

113.0.2. Second Derivative Test

If is continuous near a critical point , and , then:

Example

If , then is a local minimum (concave up)

Suppose

Step 1. Find critical points

Solve

So, is the only critical point

Step 2. Evaluate the second derivative at

Since the second derivative is positive, the function is concave up near

Step 3. Interpretation

Example

If , then is a local maximum (concave down)

Suppose

Step 1. Find critical points

Solve

So, is the only critical point

Step 2. Evaluate the second derivative at

Since the second derivative is negative, the function is concave down near

Step 3. Interpretation

Example

If , the test is inconclusive — use the first derivative test or other methods

Suppose

Step 1. Find critical points

Solve

So, is the only critical point

Step 2. Evaluate the second derivative at

Since the second derivative is 0, the test in inconclusive

Step 3. Use another method

Test sign of around

  • Left:
  • Right:

Step 4. Interpretation

Since does not change sign at ,

but rather a point of inflection