Solved: "Find the absolute maximum value and the absolute..."

High School
Algebra
Algebra I
Piecewise-Defined Functions

villane0

Answered question

2021-12-03

Find the absolute maximum value and the absolute minimum value of the function.

h (t) = t^{3} - 9 t^{2} o n [4, 8]

Answer & Explanation

Mary Darby

Beginner2021-12-04Added 11 answers

Step 1
Given,
$h (t) = t^{3} - 9 t^{2} o n [4, 8]$
Step 2
Now,
$∴ h (t) = t^{3} - 9 t^{2}$
Differentiating w.r.t t, we have
$\Rightarrow h^{'} (t) = \frac{d}{d t} (t^{3} - 9 t^{2})$
$\Rightarrow h^{'} (t) = \frac{d}{d t} (t^{3}) - 9 \frac{d}{d t} (t^{2})$
$\Rightarrow h^{'} (t) = 3 t^{2} - 18 t$
For critical points: h'(t)=0
$\Rightarrow 3 t^{2} - 18 t = 0$
$\Rightarrow 3 t (t - 6) = 0$
either
$\Rightarrow t = 0 or \Rightarrow t = 6$
Step 3
At t=0; h(0)=0
At $t = 4; h (4) = 4^{3} - 9 {(4)}^{3}$
$\Rightarrow h (4) = - 512$
At $t = 6; h (6) = 6^{3} - 9 {(6)}^{3}$
$\Rightarrow h (6) = - 1728$
At $t = 8; h (8) = 8^{3} - 9 {(8)}^{3}$
$\Rightarrow h (8) = - 4096$
$∴$ The absolute maximum is at x=0, and the value is 0.
$∴$ The absolute minimum is at x=8, and the value is -4096.

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