Answer to Question #103220 in Calculus for VV

Question

FIND THE MAXIMUM AND THE MINIMUM VALUES OF
3X^4 - 2X^3 - 6X^2 + 6X + 1 IN THE INTERVAL [0,2] IF THEY EXIST

Answer 1

FIND THE MAXIMUM AND THE MINIMUM VALUES OF

3 X ⁴- 2 X ³ - 6 X ² + 6 X + 1 IN THE INTERVAL [0,2] IF THEY EXIST

f(x)=3 X ⁴ - 2 X ³ - 6 X ²+ 6 X + 1

f'(x)=12 x ³-6 x ²-12 x+6

12 x ³-6 x ²-12 x+6=0

6(2 x ³-x ²-2 x+1)=0

by finding synthetic division

the roots are

x=1,-1,1/2

critical points are 0,1/2,1,2

f(1)=3-2-6+6+1

f(1)=2

f(1/2)=3(1/2) ⁴-2(1/2)³-6(1/2)²+6(1/2)+1

=39/16

f(0)=1

f(2)=3(2 ⁴)-2(x ³)-6(2²)+6(2)+1

=21

minimum is at (0,1),minimum is at (1,2), maximum is at (1/2,39/16), maximum is at (2,21).

Absolute maximum=21.

Absolute minimum=1,

local minima=2,

local maxima=39/16,

the domain is (- infinity ,+ infinity).