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Let be a twice-differentiable function such that and is continuous over an open interval containing Suppose Since is continuous over for all ( (Figure)). Here we examine how the second derivative test can be used to determine whether a function has a local extremum at a critical point. However, a function need not have a local extrema at a critical point. We know that if a continuous function has a local extrema, it must occur at a critical point. Using the second derivative can sometimes be a simpler method than using the first derivative. The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. This result is known as the first derivative test.

Therefore, to test whether a function has a local extremum at a critical point we must determine the sign of to the left and right of.

The function has a local extremum at the critical point if and only if the derivative switches sign as increases through.

If a continuous function has a local extremum, it must occur at a critical point.

Using (Figure), we summarize the main results regarding local extrema. The function has four critical points: The function has local maxima at and and a local minimum at The function does not have a local extremum at The sign of changes at all local extrema. Then the function decreases and it is marked that f’ > 0.” width=”867″ height=”429″> Figure 2. The function increases some more to d (so f’ > 0), which is the global maximum. The function has an inversion point at c, and it is marked f’(c) = 0. Then, f decreases from x = a to x = b (so f’ 0. We show that if has a local extremum at a critical point, then the sign of switches as increases through that point.Ġ. In (Figure), we show that if a continuous function has a local extremum, it must occur at a critical point, but a function may not have a local extremum at a critical point. The critical points are candidates for local extrema only. Note that need not have a local extrema at a critical point. Recall that such points are called critical points of Consequently, to locate local extrema for a function we look for points in the domain of such that or is undefined.

Both functions are increasing over the interval At each point the derivative Both functions are decreasing over the interval At each point the derivativeĪ continuous function has a local maximum at point if and only if switches from increasing to decreasing at point Similarly, has a local minimum at if and only if switches from decreasing to increasing at If is a continuous function over an interval containing and differentiable over except possibly at the only way can switch from increasing to decreasing (or vice versa) at point is if changes sign as increases through If is differentiable at the only way that can change sign as increases through is if Therefore, for a function that is continuous over an interval containing and differentiable over except possibly at the only way can switch from increasing to decreasing (or vice versa) is if or is undefined. At two points the derivative is taken and it is noted that at both f’ Figure 1. Figure c shows a function decreasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f’ > 0.

Figure b shows a function increasing concavely from (a, f(a)) to (b, f(b)). In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.Ġ. For example, has a critical point at since is zero at but does not have a local extremum at Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. State the second derivative test for local extrema.Įarlier in this chapter we stated that if a function has a local extremum at a point then must be a critical point of However, a function is not guaranteed to have a local extremum at a critical point.Explain the relationship between a function and its first and second derivatives.Explain the concavity test for a function over an open interval.Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph.State the first derivative test for critical points.

Explain how the sign of the first derivative affects the shape of a function’s graph.