If ${𝑓}^{\prime }\left(𝑥\right)$ changes from negative to positive at $𝑐$, then $𝑓\left(𝑐\right)$ is a local minimum

Suppose:

Step 1. Find critical point

Solve ${𝑓}^{\prime }\left(𝑥\right)=0$:

So, $𝑥=0$ is the only critical point

Step 2. Test sign of ${𝑓}^{\prime }\left(𝑥\right)$ around $𝑥=0$

• Left: $𝑥=-1$

• Right: $𝑥=1$:

Step 3. Interpretation

Since ${𝑓}^{\prime }\left(𝑥\right)$ changes from negativetopositive at $𝑥=0$,

If ${𝑓}^{\prime }\left(𝑥\right)$ changes from positive to negative at $𝑐$, then $𝑓\left(𝑐\right)$ is a local maximum

Suppose:

Step 1. Find critical point

Solve ${𝑓}^{\prime }\left(𝑥\right)=0$

So, $𝑥=0$ is the only critical point

Step 2. Test sign of ${𝑓}^{\prime }\left(𝑥\right)$ around $𝑥=0$

• Left: $𝑥=-1$

• Right: $𝑥=1$

Step 3. Interpretation

Since ${𝑓}^{\prime }\left(𝑥\right)$ changes from positivetonegative at $𝑥=0$,

If ${𝑓}^{\prime }\left(𝑥\right)$does not change sign, $𝑓\left(𝑐\right)$ is not a local extremum

Suppose

Step 1. Find critical point

Solve ${𝑓}^{\prime }\left(𝑥\right)=0$

So, $𝑥=0$ is the only critical point

Step 2. Test sign of ${𝑓}^{\prime }\left(𝑥\right)$ around $𝑥=0$

• Left: $𝑥=-1$

• Right: $𝑥=1$

Step 3. Interpretation

Since ${𝑓}^{\prime }\left(𝑥\right)$ does not change sign at $𝑥=0$,

but rather a point of inflection

If ${𝑓}^{″}\left(𝑐\right)>0$, then $𝑓\left(𝑐\right)$ is a local minimum (concave up)

Suppose

Step 1. Find critical points

Solve ${𝑓}^{\prime }\left(𝑥\right)=0$

So, $𝑥=0$ is the only critical point

Step 2. Evaluate the second derivative at $𝑥=0$

Since the second derivative is positive, the function is concave up near $𝑥=0$

Step 3. Interpretation

If ${𝑓}^{″}\left(𝑐\right)<0$, then $𝑓\left(𝑐\right)$ is a local maximum (concave down)

Suppose

Step 1. Find critical points

Solve ${𝑓}^{\prime }\left(𝑥\right)=0$

So, $𝑥=0$ is the only critical point

Step 2. Evaluate the second derivative at $𝑥=0$

Since the second derivative is negative, the function is concave down near $𝑥=0$

Step 3. Interpretation

If ${𝑓}^{″}\left(𝑐\right)=0$, the test is inconclusive — use the first derivative test or other methods

Suppose

Step 1. Find critical points

Solve ${𝑓}^{\prime }\left(𝑥\right)=0$

So, $𝑥=0$ is the only critical point

Step 2. Evaluate the second derivative at $𝑥=0$

Since the second derivative is 0, the test in inconclusive

Step 3. Use another method

Test sign of ${𝑓}^{\prime }\left(𝑥\right)$ around $𝑥=0$

• Left: $𝑥=-1$

• Right: $𝑥=1$

Step 4. Interpretation

Since ${𝑓}^{\prime }\left(𝑥\right)$ does not change sign at $𝑥=0$,

but rather a point of inflection