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 \(f\) 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 \(f\) 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 (three situations discussed above)

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

the local extrema.