SOLUTION: Perform a first derivative test on the function f(x)= 2x^3+3x^2-120x+2; [-5,7]. a. Locate the critical points of the given function. b. Use the first derivative test to locate th

Algebra ->  Test -> SOLUTION: Perform a first derivative test on the function f(x)= 2x^3+3x^2-120x+2; [-5,7]. a. Locate the critical points of the given function. b. Use the first derivative test to locate th      Log On


   

Question 1155974: Perform a first derivative test on the function f(x)= 2x^3+3x^2-120x+2; [-5,7].
a. Locate the critical points of the given function.
b. Use the first derivative test to locate the local maximum and minimum values.
c. Identify the absolute minimum and maximum values of the function on the given interval(if they exist).

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=+2x%5E3%2B3x%5E2-120x%2B2
[-5,7]

a. Locate the critical points of the given function.

find first derivative

f'+%28x%29+=6x%5E2%2B6x-120
set f'+%28x%29+=0

6x%5E2%2B6x-120=0
6%28x%5E2%2Bx-20%29=0
will be zero if
x%5E2%2Bx-20=0.....factor
x%5E2-4x%2B5x-20=0
%28x%5E2-4x%29%2B%285x-20%29=0
x%28x-4%29%2B5%28x-4%29=0
%28x+-+4%29+%28x+%2B+5%29+=+0

=> extreme points are at

x=4 and x=-5

use second derivative test to determine where is max and where is min
f''%28x%29+=12x%2B6

f''%28-5%29+=12%28-5%29%2B6=-54 => negative, the absolute maximum is at+-5

f''%284%29+=12%284%29%2B6=54 => positive, the absolute minimum is at 4


+graph%28+600%2C+600%2C+-15%2C+15%2C+-500%2C+500%2C+2x%5E3%2B3x%5E2-120x%2B2%29+