Question

Find the exact location of all the relative and absolute extrema of the function. HINT [See Examples 1 and 2.] (Order your answers from smallest to largest t.) h(t) = 8t3 − 24t2 with domain [−1, +[infinity])

Answer 1

Answer:

• (-1, -32) absolute minimum
• (0, 0) relative maximum
• (2, -32) absolute minimum
• (+∞, +∞) absolute maximum (or "no absolute maximum")

Step-by-step explanation:

There will be extremes at the ends of the domain interval, and at turning points where the first derivative is zero.

The derivative is ...

  h'(t) = 24t^2 -48t = 24t(t -2)

This has zeros at t=0 and t=2, so that is where extremes will be located.

We can determine relative and absolute extrema by evaluating the function at the interval ends and at the turning points.

  h(-1) = 8(-1)²(-1-3) = -32

  h(0) = 8(0)(0-3) = 0

  h(2) = 8(2²)(2 -3) = -32

  h(∞) = 8(∞)³ = ∞

The absolute minimum is -32, found at t=-1 and at t=2. The absolute maximum is ∞, found at t→∞. The relative maximum is 0, found at t=0.

The extrema are ...

• (-1, -32) absolute minimum
• (0, 0) relative maximum
• (2, -32) absolute minimum
• (+∞, +∞) absolute maximum

_____

Normally, we would not list (∞, ∞) as being an absolute maximum, because it is not a specific value at a specific point. Rather, we might say there is no absolute maximum.