Question

In Exercises, you are given \(f^{\prime}\). Find the intervals on which (a) \(f^{\prime}(x)\) is increasing or decreasing and (b) the graph of \(f\) is concave upward or concave downward. (c) Find the relative extrema and inflection points of \(f\). (d) Then sketch a graph of \(f\). $$ f^{\prime}(x)=-x^{2}+2 x-1 $$

Short answer

We first identify critical points which are \(\frac{1 - \sqrt{3}}{2}\) and \(\frac{1 + \sqrt{3}}{2}\). We then test intervals around these critical points and find that \(f'(x)\) is decreasing on (-∞, \(\frac{1 - \sqrt{3}}{2}\)) ∪ (\(\frac{1 + \sqrt{3}}{2}\), ∞) and increasing on (\(\frac{1 - \sqrt{3}}{2}\), \(\frac{1 + \sqrt{3}}{2}\)). Next, we find the second derivative to be -2x+2. Setting it to zero yields the inflection point x=1. The function is concave upwards for x<1 and concave downwards for x>1. Finally, we sketch the plot of f(x) based on these findings.

Step-by-step solution

Find where \(f^{\prime}(x)\) is increasing or decreasing

The derivative \(f^{\prime}(x) = -x^{2} + 2x - 1\). We set this equal to 0 to find the critical points. Solving \(-x^{2} + 2x - 1 = 0\) will give us the required critical points. It is important to analyze the intervals around these critical points to see where the derivative is increasing or decreasing.

Analyze the concavity of \(f\)

We can find the intervals in which the graph of \(f\) is concave upward or downward by finding the second derivative \(f^{\prime\prime}(x)\) of the given function. The function is concave upward where \(f^{\prime\prime}(x) > 0\) and concave downward where \(f^{\prime\prime}(x) < 0 \).

Find the relative extrema and inflection points of \(f\)

Relative extrema occur where the derivative \(f^{\prime}(x)\) changes sign (from plus to minus or vice versa), and are considered to be either relative minimums or maximums. Inflection points, on the other hand, occur where the second derivative \(f^{\prime\prime}(x)\) changes sign.

Sketch the graph of \(f\)

When drawing the graph, we need to observe the sign of the first and second derivatives in each interval - this determines the increasing/decreasing nature and the concavity of the function, respectively. The plot should show the inflection points and the relative extrema that we identified in the previous steps.

Key concepts

Critical Points

Critical points on a graph are essentially where the function changes behavior. These are the points where the function's derivative which represents the rate of change of the function equals zero or doesn't exist. When we analyze the function f'(x) = -x^2 + 2x - 1, finding the critical points involves solving f'(x) = 0. The solutions to this equation tell us where the slope of the tangent line to the function's graph is zero, which typically indicates a local maximum or minimum.

It's much like hiking to a hilltop and finding a flat spot where you're no longer going up or down — reaching a critical point. Understanding these points is crucial in determining where the function attains its highest or lowest values on certain intervals, thereby shaping the entire curve of the graph.

First Derivative Test

The first derivative test is a handy tool for analyzing the behavior of functions around their critical points. Once we've found the critical points, we can pick values from intervals around these points to see if the derivative f'(x) is positive or negative in those regions. This tells us whether the function f(x) is increasing or decreasing there.

For our function with f'(x) = -x^2 + 2x - 1, if f'(x) changes from positive to negative as we pass through a critical point, the function has a local maximum at that point. If the change is from negative to positive, we're looking at a local minimum. This helps us sketch part of the function's graph, bringing us closer to understanding the full picture of how the graph of f behaves.

Second Derivative Test

The second derivative test is similar to the first derivative test but focuses on the concavity of the function f and its inflection points where the concavity changes. By taking the derivative of f'(x), we obtain the second derivative f''(x). If f''(x) > 0 for certain values of x, the graph is concave upward like a cup. Conversely, if f''(x) < 0, the graph is concave downward like a cap.

The points where the second derivative changes its sign are the inflection points. These points are significant because they indicate a change in the curve's direction, showing where the graph switches from being 'smiley' (concave up) to 'frowny' (concave down) or vice versa. In our function, we would calculate f''(x) and test intervals to find where these changes occur.

Sketching Graphs of Functions

When it's time to sketch the graph of a function, the combination of critical points, extrema, and inflection points gives us a roadmap. We've already determined where the function increases or decreases and where it switches concavity - this foundational understanding informs our sketch.

Starting with plotting the critical points on a graph, we then show the intervals where the function is increasing or decreasing as determined by the first derivative test. Next, we add points of inflection from the second derivative test and illustrate the concave up and down segments. By connecting all these points with smooth curves that respect the increasing/decreasing and concavity information, we get a good approximation of the function's graph.

For the function f'(x) = -x^2 + 2x - 1, after working through the prior steps, the sketch would reveal the overall shape of f, including any peaks, valleys, and where it bends, just like a map would show the terrain of a hiking trail.