Question

Suppose a continuous function \(f\) is concave up on \((-\infty, 0)\) and \((0, \infty) .\) Assume \(f\) has a local maximum at \(x=0 .\) What, if anything, do you know about \(f^{\prime}(0) ?\) Explain with an illustration.

Short answer

Answer: The value of the derivative f'(0) is 0. A visual explanation is provided by the graph of a concave up function like f(x)=x^3. The local maximum occurs at x=0, where the tangent line to the curve of the function is horizontal, indicating a derivative value of zero.

Step-by-step solution

Concavity and its relation to the function

A function is said to be concave up on an interval if its rate of increase is increasing, that is, its second derivative is positive on that interval. In this case, \(f''(x) > 0\) for \(x < 0\) and \(x > 0\).

Local maximum and its relation to the function

A function is said to have a local maximum at a point if the value of the function at that point is greater than the values of the function in the neighborhood of that point. In other words, there is an open interval around that point where the function value at that point is the highest. In this case, we have a local maximum at \(x=0\).

Derivative at local maximum

According to Fermat's theorem, if a function has a local maximum (or minimum) at some point \(c\) and the derivative exists at that point, then the derivative of the function at that point must be equal to zero. That is, \(f'(c)=0\). In our case, \(c=0\), so we expect \(f'(0)=0\).

Confirming the derivative value

Because \(f\) is concave up on \((-\infty, 0)\) and \((0, \infty)\), the function has a "U" shape. Thus, when moving from left to right, the slope of the tangent lines to the graph of \(f\) will gradually increase from negative values to positive values, passing through zero at the local maximum, \(x=0\). Therefore, the function's derivative, \(f'(0)=0\).

Illustration

To provide a visual explanation, draw the graph of a concave up function like \(f(x)=x^3\). Notice that the local maximum occurs at \(x=0\), where the tangent line to the curve of the function is horizontal, indicating a derivative value of zero.