20 Properties of the Derivative the Derivative of a Function Can Be Used

Math 321 - March 01, 2021 20 Properties of the derivative The derivative of a function can be used to solve optimization problems, as it gives an easy way of identifying extrema of a function. Theorem 20.1 (Interior Extremum Theorem). Let f be differentiable on an open interval (a;b). If f attains a maximum value at some point c2(a;b), that is, f(c)≥f(x) for all x2(a;b), then f0(c)=0. The same is true if f(c) is a minimum value. The proof of the last theorem uses the fact that a function must stagnate at an extremum point, i.e. it can be neither increasing (f0(c)≤0) nor decreasing (f0(c)≥0). As we saw before, a differentiable function must be continuous. On the other hand, the derivative function itself may not be continuous, as the example f(x)=x2sin(1=x) shows. However, the derivative function cannot have removable or jump discontinuities (i.e. all the discontinuities of a derivative function must be essential), as the next statement implies. Theorem 20.2 (Darboux's Theorem). If f is differentiable on an interval [a;b], and α satisfies f0(a)<α<f0(b), or f0(b)<α<f0(a), then there exists a point c2(a;b) where f0(c)=α. The above theorem says that the intermediate value property must hold for derivative functions. The proof follows from applying the Interior Extremum Theorem to the function g(x)=f(x)−αx. Combining the Interior Extremum Theorem with Extreme Value Theorem, one has the following. Theorem 20.3 (Rolle's Theorem). Let f :[a;b] ! R be continuous on [a;b] and differentiable on (a;b). If f(a)=f(b), then there exists a point c2(a;b), where f0(c)=0. Indeed, the proof follows from the observation that both extrema of f cannot be attained at the endpoints, unless f is a constant function. Rolle's theorem can be generalized to cases where the values of the function are not the same at the endpoints. In that case, the theorem would imply the existence of a point, where the derivative equals the slope of the line connecting the endpoints of the graph of f over [a;b]. Theorem 20.4 (Mean Value Theorem). If f :[a;b]!R is continuous on [a;b] and differentiable on (a;b), then there exists a point c2(a;b) where f(b)−f(a) f0(c)= : b−a The proof follows from an application of Rolle's theorem to the function g(x) = f(x)−[(f(b)−f(a))=(b− a)](x−a). The Mean Value Theorem has many applications, one of which is the following. Corollary 20.5. If g : A ! R is differentiable on an interval A, and satisfies g0(x) = 0 for all x 2 A, then g(x)≡k for some constant k2R. Applying the last statement to a difference of two functions gives. Corollary 20.6. If f and g are differentiable functions on an interval A and satisfy g0(x)=f0(x) for all x2A, then f(x)≡g(x)+k for some constant k2R..

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