Resolved: "Find absolute maximum and the absolute minimum values,..."

High School
Algebra
Algebra I
Piecewise-Defined Functions

grasaladae5

Answered question

2021-12-04

Find absolute maximum and the absolute minimum values, if any, of the function on [-3,2]

h (x) = x^{3} + 3 x^{2} + 1

Answer & Explanation

Richard Cheatham

Beginner2021-12-05Added 16 answers

Step 1
To find the absolute maximum and minimum on [-3, 2]
Step 2

h (x) = x^{3} + 3 x^{2} + 1

on [-3,2]
First find the critical points

h^{'} (x) = 3 x^{2} + 6 x

Put h'(x)=0

3 x^{2} + 6 x = 0

3x(x+2)=0
x=0, x=-2
Now we have four points on which we have to check absolute minimum and absolute maximum and points are x=-3, -2, 0, 2
Put x=-3 in f(x)

f (- 3) = {(- 3)}^{3} + 3 {(- 3)}^{2} + 1 = 1

Put x=-2,

f (- 2) = {(- 2)}^{3} + 3 \times {(- 2)}^{2} + 1 = 5

Put x=0
f(0)=0+0+1=1
Put x=2 then

f (2) = 2^{3} + 3 \times 2^{2} + 1 = 21

So, now we can see it is minimum at x=-3 and minimum value is 1
And maximum at x=2, and maximum value is 21

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