Question

True or False If \(f^{\prime}(c)=0\) and \(f^{\prime \prime}(c)<0,\) then \(f(c)\) is a local maximum. Justify your answer.

Short answer

True. If \(f'(c) = 0\) and \(f''(c) < 0\), then \(f(c)\) is a local maximum.

Step-by-step solution

Analyzing the condition of the first derivative

The first derivative of the function \(f\) at point \(c\) is given as zero. This means \(f'(c) = 0\). It implies that the point \(c\) is a critical point and the function doesn't increase or decrease at this point, and potentially it could be a maximum or minimum.

Analyzing the condition of the second derivative

The second derivative of the function \(f\) at point \(c\) is given as less than zero, it means \(f''(c) < 0\). The concavity of a function at a point is determined by the second derivative. If the second derivative is positive, the function is concave up which indicates a local minimum. But if the second derivative is negative, the function is concave down, which indicates a local maximum.

Final analysis and conclusion

Taking both the conditions together, it can be concluded that if \(f'(c) = 0\) and \(f''(c) < 0\), then the function \(f\) has a local maximum at \(c\). Its because at point \(c\), the function doesn't increase or decrease (due to zero first derivative) and has a concave down shape (due to negative second derivative) which characterizes a local maximum.

Key concepts

First Derivative Test

When analyzing the behavior of a graph of a function, the first derivative test is a prime tool for identifying local extrema (minimums and maximums). The test uses the first derivative, or slope, of the function at a point to determine whether the function is increasing or decreasing near that point. More precisely, if the function changes from increasing to decreasing at a point, this point is a local maximum. Conversely, if the function changes from decreasing to increasing, that point is considered a local minimum. The first derivative test also introduces the concept of a critical point, which occurs when the first derivative is zero or undefined. This is a potential indicator of a local extremum, but further analysis is needed to confirm it.

To clearly understand, when you plot the curve of the function, the slope of the tangent at any point describes how steeply the curve ascends or descends at that point. At a local maximum, the slope changes from positive (uphill) to negative (downhill), indicating the top of a hill on the graph.

Second Derivative Test

The second derivative test is another useful method to determine local maximums (or peaks) and minimums (or valleys). It involves the second derivative of the function, which informs us about the concavity of the curve. If the second derivative is positive at a critical point where the first derivative is zero, the graph is concave up, resembling the shape of a U, and the point is a local minimum. If the second derivative is negative, the graph is concave down, like an upside-down U, signifying a local maximum. A critical point where the function has a zero first derivative but a negative second derivative reveals a hilltop on the graph's landscape; the curve is bending down on either side of this point.

For example, when you are standing at the highest point on a bridge, the road curves downwards in both directions from your viewpoint. This is analogous to a negative second derivative at the point where you are standing - representing a local maximum on the graph of the bridge's profile.

Critical Point

A critical point is where the first derivative of a function is either zero or undefined. These points are crucial in the investigation of a function’s graph because they are the candidates for local extrema and points of inflection. They are also the spots where the tangent to the curve is horizontal or vertical. However, not every critical point is a local maximum or minimum. To determine the nature of the extremum at a critical point, we must perform tests like the first and second derivative tests.

Imagine you are exploring a landscape with hills and valleys. A critical point would be any spot where the path you walk becomes flat or takes a sharp turn vertically. Importantly, just because the path is flat does not mean you're on top of a hill; it could be a plateau or the bottom of a valley. The additional derivative tests help to distinguish between these possibilities.

Concavity

Concavity describes the direction a curve bends and is determined by the sign of the second derivative of a function. If the second derivative is positive, the curve is concave up, meaning it holds up like a cup, and we can expect local minimums to emerge at critical points. If the second derivative is negative, the curve is concave down, like an arch, suggesting the presence of local maximums at critical points. The concavity of a function greatly aids in graphing the function and understanding its behavior between critical points.

Analogously, consider holding a piece of string with both hands and letting it droop. The string demonstrates concave up behavior, curving upwards. In contrast, if you push the middle of the string up while still holding the ends, it will mimic a concave down shape, curving downwards. These visual cues closely resemble how a function's graph would behave in regards to its concavity.