When determining the sign of \(f^\prime\) is difficult, we can use another test for local maximum and minimum values. This test is based on the geometrical observation that when the function has a horizontal tangent at \(c\), if the function is concave down, the function has a local maximum at \(c\), and if it is concave up, it has a local minimum (see Figure 1)

**Theorem 1. (Second Derivative Test)**Suppose \(f^\prime(c)=0\).*(a) If \(f^{\prime\prime}(c)>0\), then \(f\) has a local minimum at \(c\).*

*(b) If \(f^{\prime\prime}(c)<0\), then \(f\) has a local maximum at \(c\).*

#### Show the proof …

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- For sufficiently small \(\Delta x<0\), \(f^\prime(c+\Delta x)\) must be negative. This means (Increasing/Decreasing Theorem) \(f\) is decreasing \(\searrow\) in some interval to the left of \(c\).
- For sufficiently small \(\Delta x>0\), \(f^\prime(c+\Delta x)\) must be positive. This means (Increasing/Decreasing Theorem) \(f\) is increasing \(\nearrow\) in some interval to the right of \(c\).

Therefore, \(f\) has a local minimum at \(c\). The proof of part (b) is analogous.

- Recall that if \(g\) is continuous at \(c\) and \(g(c)\neq0\), then \(g(x)\) has the same sign as \(g(c)\) for \(x\) sufficiently close to \(c\). Therefore, if \(f^{\prime\prime}\) is continuous at \(c\) and \(f^{\prime\prime}(c)\neq0\), we can say that the function is concave up near \(c\) if \(f^{\prime\prime}(c)>0\) and is concave down near \(c\) if \(f^{\prime\prime}(c)<0\).
- There are three situations where the Second Derivative Test is
**inconclusive**:- \(f^\prime(c)=f^{\prime\prime}(c)=0\)
- \(f^\prime(c)=0\) and \(f^{\prime\prime}(c)\) does not exist.
- \(f^\prime(c)\) does not exist.

In these cases, \(c\) may be a local minimum point, a local maximum point, or neither as shown (Figure 2) by the functions \[f(x)=x^{4},\quad f(x)=-x^{4},\quad f(x)=x^{3}.\] For these functions, \(f^\prime(0)=f^{\prime\prime}(0)=0\), but \(x=0\) is a point of local minimum for \(f(x)=x^{4}\), a point of local maximum for \(f(x)=-x^{4}\) and neither a local minimum nor maximum point for \(f(x)=x^{3}\).

- Whenever the Second Derivative Test is inconclusive (as in the three situations discussed above)

or when the second derivative is tedious to find, use the First Derivative Test to find

the local extrema.