Check the solution to "Find the critical points of the following functions..."

sanuluy

Answered question

2020-11-01

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

f (x) = x^{3} - 6 x^{2} on [- 1, 5]

Answer & Explanation

Dora

Skilled2020-11-02Added 98 answers

Step 1 Let us find the critical points , absolute maximum and minimum of given function Given: $f (x) = x^{3} - 6 x^{2} on [- 1, 5]$

Step 2 $f (x) = x^{3} - 6 x^{2}$
$f^{'} (x) = 3 x^{2} - 12 x$
$0 = 3 x (x - 4)$
$3 x = 0, x - 4 = 0$
$x = 0, x = 4$ Critical point $= 0, 4$

Step 3 To find absolute maximum and absolute minimum: $f (x) = x^{3} - 6 x^{2}$
$x = - 1$
$f (- 1) = {(- 1)}^{3} - 6 {(- 1)}^{2}$
$f (- 1) = - 1 - 6$
$f (- 1) = - 7$
$x = 0$
$f (0) = 0 - 6 (0)$
$f (0) = 0$
$x = 4$
$f (4) = 4^{3} - 6 {(4)}^{2}$
$f (4) = 64 - 6 (16)$
$f (4) = 64 - 96$
$f (4) = - 32$
$x = 5$
$f (5) = 5^{3} - 6 {(5)}^{2}$
$f (5) = 125 - 150$
$f (5) = - 25$ Absolute maximum at y=0, x=0 Absolute minimum at y=-32, x=5

Step 4 Answer: Critical points at $x = 0, 4$ Absolute maximum at $x = 0$ Absolute minimum at $x = 5$

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