Which report of the absolute maximum on [0, 6] is complete and correct?

The absolute maximum value is 36, and it occurs at x = 6.

The absolute maximum value is 20, and it occurs at x = 2.

The absolute maximum value is 6, and it occurs at x = 36.

The absolute maximum value is 36.

Which report of the absolute minimum on [0, 6] is correct?

The absolute minimum value is 0, and it occurs at x = 0.

The absolute minimum value is 16, and it occurs at x = 4.

There is no absolute minimum, because the left end is not a hump or a valley.

A company's profit is P(t) = 12t - 2t^2 thousand dollars, t months after a launch, for 0 \le t \le 5. Which report of the maximum profit is complete and correct?

The maximum profit is 18 thousand dollars, in month 3.

The maximum profit is 3 thousand dollars, in month 18.

The maximum profit is 10 thousand dollars, in month 5.

The maximum profit is 18 thousand dollars.

Topic 5.5

Report absolute extrema in full

Builds on comparing function values at the candidates. 6 minutes.

The temperature in a greenhouse

The graph shows the temperature in a greenhouse, \(T(h) = -h^2 + 8h + 4\) degrees Celsius, \(h\) hours after a heater starts, for \(0 \le h \le 6\).

Move the slider to read the temperature at each hour. Then press the button to go to the highest point. The dashed line across to the vertical axis shows its height, 20, the absolute maximum value. The dashed line down to the horizontal axis shows its input, 4, where the maximum occurs.

A full answer gives both numbers with their units: the highest temperature is 20 degrees Celsius, and it occurs 4 hours after the start.

A value and an input

This curve is continuous on \([a, b]\). The large dots mark its highest and lowest points, and each is one point with two coordinates.

The dashed lines carry the highest point to the axes. Its height \(M\) is the absolute maximum value, and its input \(c\) is where the maximum occurs, so \(f(c) = M\). The lowest point gives the absolute minimum value \(m\) and its input \(d\).

Move the Tilt slider. The curve changes and the highest point moves from one top to the other, but it stays one point with a height and an input.

Notation

If \(M\) is the absolute maximum value, then \(M = f(c)\) for some \(c\) in the interval, and \(f(x) \le M\) for every \(x\) in it.
\[\begin{gathered} f(c) = \cy{M} \\ f(x) \le \cy{M} \end{gathered}\]
In words, the absolute maximum value is \(M\), and it occurs at \(x = c\).
For the absolute minimum value \(m\), the inequality turns around.
\[\begin{gathered} f(d) = \cy{m} \\ f(x) \ge \cy{m} \end{gathered}\]
If the question asks how large, give the value. If it asks where or when, give the input. If it asks for both, give both, with their units.

When it fails

The value \(M\) is one number, but the input can be more than one. Press the toggle to set the tilt to 0, so the two tops have the same height.

The dashed line at the height of the tops touches the graph twice. The maximum value occurs at two inputs, so a report with a single input is incomplete.

Example 1: A medicine in the blood

The concentration of a medicine is \(C(t) = 4te^{-t/2}\) milligrams per liter, \(t\) hours after a dose, for \(0 \le t \le 6\). Find the highest and lowest concentrations and the times when they occur.

The product rule gives \(C'\). The factor \(e^{-t/2}\) is never 0, so \(C' = 0\) only where \(4 - 2t = 0\), at \(t = 2\).
\[\begin{gathered} C'(t) = (4 - 2t)e^{-t/2} \\ C'(t) = 0 \quad\Rightarrow\quad t = 2 \end{gathered}\]
Evaluate \(C\) at the candidates: the ends 0 and 6, and \(t = 2\).
\[\begin{gathered} C(0) = 0 \\ C(2) = \tfrac{8}{e} \\ C(6) = 24e^{-3} \end{gathered}\]
As a check, \(8/e \approx 2.943\) and \(24e^{-3} \approx 1.195\), so \(8/e\) is the largest value and 0 the smallest.
The highest concentration is \(8/e\) milligrams per liter, at \(t = 2\) hours. The lowest is 0 milligrams per liter, at \(t = 0\).

Example 2: A tank and a pump

A pump changes the volume of water in a tank by \(V(t) = t^3 - 6t^2\) liters, \(t\) hours after it starts, for \(0 \le t \le 5\). Find the lowest and highest values of \(V\) and where they occur.

Factor the derivative. The zeros \(t = 0\) and \(t = 4\) both lie in the interval.
\[V'(t) = 3t^2 - 12t = 3t(t - 4)\]
Evaluate \(V\) at the candidates 0, 4 and 5.
\[\begin{gathered} V(0) = 0 \\ V(4) = -32 \\ V(5) = -25 \end{gathered}\]
A tempting wrong move: The lowest point is at \(t = 4\), so the lowest value of \(V\) is 4.
\[\text{minimum} = 4\]
The dashed line at height 4 lies above the whole graph, so \(V\) never reaches 4. The number 4 is a place on the horizontal axis, not a height, and the point there is at height \(-32\).
The lowest value is the height of that point, \(-32\) liters, and it occurs at \(t = 4\) hours. The highest value is 0 liters, at \(t = 0\).

Practice

The elevation of a trail above its start is \(g(x) = 2x^3 - 15x^2 + 24x\) meters, \(x\) kilometers from the start, for \(0 \le x \le 5\). Find the highest and lowest elevations and where they occur.

Drag the point to the top and to the bottom of the graph. Report the height of each, in meters, and its place on the horizontal axis, in kilometers.

Solution

The derivative is 0 at \(x = 1\) and \(x = 4\), and both lie in the interval.
\[\begin{aligned} g'(x) &= 6x^2 - 30x + 24 \\ &= 6(x - 1)(x - 4) \end{aligned}\]
Evaluate \(g\) at 0, 1, 4 and 5.
\[\begin{gathered} g(0) = 0, \quad g(1) = 11 \\ g(4) = -16, \quad g(5) = -5 \end{gathered}\]
The highest elevation is 11 meters, at 1 kilometer. The lowest elevation is \(-16\) meters, at 4 kilometers.

Questions

These questions use the graph of \(r(x) = x^3 - 9x^2 + 24x\) on \([0, 6]\), which has a hump at \(x = 2\) and a valley at \(x = 4\).

Which report of the absolute maximum on \([0, 6]\) is complete and correct?
- The absolute maximum value is 36, and it occurs at \(x = 6\).
- The absolute maximum value is 20, and it occurs at \(x = 2\).
- The absolute maximum value is 6, and it occurs at \(x = 36\).
- The absolute maximum value is 36.
Answer
The absolute maximum value is 36, and it occurs at \(x = 6\).
Which report of the absolute minimum on \([0, 6]\) is correct?
- The absolute minimum value is 0, and it occurs at \(x = 0\).
- The absolute minimum value is 16, and it occurs at \(x = 4\).
- There is no absolute minimum, because the left end is not a hump or a valley.
Answer
The absolute minimum value is 0, and it occurs at \(x = 0\).
A company's profit is \(P(t) = 12t - 2t^2\) thousand dollars, \(t\) months after a launch, for \(0 \le t \le 5\). Which report of the maximum profit is complete and correct?
- The maximum profit is 18 thousand dollars, in month 3.
- The maximum profit is 3 thousand dollars, in month 18.
- The maximum profit is 10 thousand dollars, in month 5.
- The maximum profit is 18 thousand dollars.
Answer
The maximum profit is 18 thousand dollars, in month 3.

Summary

The highest point of the temperature graph has height 20 degrees Celsius at input 4 hours.

An absolute extreme is a point of the graph, so its full report has a value and an input, each with its unit.

If a question asks for an extreme and where it occurs, give both numbers.

This is one of five skills in topic 5.5 of Which & Why: AP Calculus.