SOLUTION: Consider the function 𝑓(𝑥)=3𝑥⁵−5𝑥³ which is defined on [-3, 3] a) Draw the graph of this function. b) Identify the optimal point(s) of this function. c) Find th

Algebra ->  Rational-functions -> SOLUTION: Consider the function 𝑓(𝑥)=3𝑥⁵−5𝑥³ which is defined on [-3, 3] a) Draw the graph of this function. b) Identify the optimal point(s) of this function. c) Find th      Log On


   



Question 1170636: Consider the function 𝑓(𝑥)=3𝑥⁵−5𝑥³ which is defined on [-3, 3]
a) Draw the graph of this function.
b) Identify the optimal point(s) of this function.
c) Find the global max and min points of this function.

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29+=+3x%5E5+-+5x%5E3 defined on [-3, 3]
Identify the optimal point(s) of this function.
Find the global max and min points of this function.
x-intercepts:
0+=+%283x%5E2+-+5%29x%5E3-> x=0 or
0+=+%283x%5E2+-+5%29 ->5+=+%283x%5E2+-+5%29 ->5%2F3+=+x%5E2 -> x=%2B-sqrt%285%2F3%29+
x-intercepts are at: (sqrt%285%2F3+%29,0) and (-sqrt%285%2F3+%29,0)
y-intercept:
f%28x%29+=+3%2A0%5E5+-+5%2A0%5E3=0
y-intercept at: (0,0)
global max :
use first derivative test

%28d%2Fdx%29%283x%5E5+-+5x%5E3%29+=5%2A3x%5E4-3%2A5x%5E2
%28d%2Fdx%29%283x%5E5+-+5x%5E3%29+=15x%5E4-15x%5E2
%28d%2Fdx%29%283x%5E5+-+5x%5E3%29+=+15x%5E2%28x%5E2+-+1%29
->%28x%5E2+-+1%29=0 ->x%5E2+=+1->x=1 or x=-1

then
f%28x%29+=+3%2A1%5E5+-+5%2A1%5E3=-2
f%28x%29+=+3%2A%28-1%29%5E5+-+5%2A%28-1%29%5E3=-3-%28-5%29=-3%2B5=2


max{ +3x%5E5+-+5x%5E3 } =+2 atx+=+-1
and
min { 3x%5E5+-+5x%5E3 } =+-2 at x+=+1
use given interval as critical points:
f%28-3%29+=+3%2A%28-3%29%5E5+-+5%2A%28-3%29%5E3=-594
min (-3,+-594 )
f%283%29+=+3%2A%283%29%5E5+-+5%2A%283%29%5E3=594
max(3,594)