Get the solution to "Find the absolute maximum and absolute minimum values..."

High School
Algebra
Algebra I
Piecewise-Defined Functions

usagirl007A

Answered question

2021-03-06

Find the absolute maximum and absolute minimum values of function on the given interval.

f (x) = 5 + 54 x - 2 x^{3}, [0, 4]

Answer & Explanation

tabuordg

Skilled2021-03-07Added 99 answers

Step 1
Given:
Given that function $f (x) = 5 + 54 x - 2 x^{3}$
The given interval is $[0, 4]$
To find:
we have to find the absolute maximum and absolute minimum
Step 2
Here $f (x) = 5 + 54 x - 2 x^{3}$ ...(1) and the interval is $[0, 4]$
Differentiate (1) w.r.t.x, we get
$\frac{d}{d x} f (x) = \frac{d}{d x} (5 + 54 x - 2 x^{3})$
$f^{'} (x) = 54 - 6 x^{2}$
set $f^{'} (x) = 0$
$\Rightarrow 54 - 6 x^{2} = 0$
$\Rightarrow 6 x^{2} = 54$
$\Rightarrow x^{2} = 9$
$\Rightarrow x = 3$ and $x = - 3$
but $x = - 3 \notin [0, 4]$
and it also contain the critical point $x = 0$
Step 3
Hence, the critical points is $x = 3$ and $x = 0$
Now, $f (x) = 5 + 54 x - 2 x^{3}$
For $x = 3$
$f (3) = 5 + 54 (3) - 2 {(3)}^{3} = 5 + 162 - 54$
$= 113$
Thus, the absolute value is maximum at $x = 3$
Now, $f (x) = 5 + 54 x - 2 x^{3}$
For $x = 0$
$f (0) = 5$
Thus, the absolute value is minimum at $x = 0$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Algebra I

Didn't find what you were looking for?