Chapter 4 | Applications of Derivatives
391
A continuous function f has a local maximum at point c if and only if f switches from increasing to decreasing at point c . Similarly, f has a local minimum at c if and only if f switches from decreasing to increasing at c . If f is a continuous function over an interval I containing c and differentiable over I , except possibly at c , the only way f can switch from increasing to decreasing (or vice versa) at point c is if f ′ changes sign as x increases through c . If f is differentiable at c , the only way that f ′. can change sign as x increases through c is if f ′( c ) =0. Therefore, for a function f that is continuous over an interval I containing c and differentiable over I , except possibly at c , the onlyway f can switch from increasing to decreasing (or vice versa) is if f ′( c ) =0 or f ′( c ) is undefined. Consequently, to locate local extrema for a function f , we look for points c in the domain of f such that f ′( c ) =0 or f ′( c ) is undefined. Recall that such points are called critical points of f . Note that f need not have a local extrema at a critical point. The critical points are candidates for local extrema only. In Figure 4.31 , we show that if a continuous function f has a local extremum, it must occur at a critical point, but a function may not have a local extremum at a critical point. We show that if f has a local extremum at a critical point, then the sign of f ′ switches as x increases through that point.
Figure 4.31 The function f has four critical points: a , b , c , and d . The function f has local maxima at a and d , and a local minimum at b . The function f does not have a local extremum at c . The sign of f ′ changes at all local extrema.
Using Figure 4.31 , we summarize the main results regarding local extrema. • If a continuous function f has a local extremum, it must occur at a critical point c . • The function has a local extremum at the critical point c if and only if the derivative f ′ switches sign as x increases through c . • Therefore, to test whether a function has a local extremum at a critical point c , we must determine the sign of f ′( x ) to the left and right of c . This result is known as the first derivative test .
Made with FlippingBook - professional solution for displaying marketing and sales documents online