Question

Graphical Analysis Consider the functions \(f\) and \(g,\) where \(f(x)=6 \sin x \cos ^{2} x \quad\) and \(\quad g(t)=\int_{0}^{t} f(x) d x\) (a) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (b) Explain why \(g\) is nonnegative. (c) Identify the points on the graph of \(g\) that correspond to the extrema of \(f\) (d) Does each of the zeros of \(f\) correspond to an extremum of \(g ?\) Explain. (e) Consider the function \(h(t)=\int_{\pi / 2}^{t} f(x) d x\). Use a graphing utility to graph \(h\). What is the relationship between \(g\) and \(h\) ? Verify your conjecture.

Short answer

Function \(f\) oscillates between positive and negative values but remains symmetric about the x-axis, resulting in a nonnegative function \(g\). The extrema points of \(f\) correspond to the points where \(g\) changes direction, while the zeros of \(f\) correspond to the extrema of \(g\). Lastly, the graph of \(h\) reveals the shift in the accumulation point from 0 to \(\pi/2\) in comparison with \(g\).

Step-by-step solution

Graphing the Functions

Use a graphing utility to graph functions \(f\) and \(g\). Observe their shapes, noting the general behavior of each function.

Nonnegativity of \(g\)

Explain why \(g\) is nonnegative. This can be done by observing the function \(f\) since \(g\) is the integral of \(f\). Since \(f\) oscillates between positive and negative but is symmetrical about the x-axis, the positive and negative areas cancel out making \(g\) nonnegative.

Identifying Extrema Points

Identify the points on the graph of \(g\) that correspond to the extrema of \(f\). These points are where \(f\) reaches its peak (positive or negative) since \(g\) is the integral of \(f\), these points would be where \(g\) changes direction and hence would be extrema of \(g\).

Analyzing the Zeros of \(f\)

Analyze whether each of the zeros of \(f\) correspond to an extremum of \(g\). Observe this by looking at where \(f\) crosses the x-axis. At these points, \(f\) stops contributing to \(g\)'s total (as it briefly becomes 0), so the slope of \(g\) is also zero. This means these points are extrema for \(g\).

Introduction and Analysis of Function \(h\)

Consider function \(h(t) = \int_{\pi/2}^t f(x) dx\). Then use a graphing utility to graph \(h\), observing the shape and features of the graph in relation to \(g\). Finally, compare the three functions \(f\), \(g\), and \(h\) to confirm conjectures regarding their relationships.