SOLUTION: F(x) =2x^2 - 3x^(4/3) a) find critical points b) find inflection points

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Question 1180983: F(x) =2x^2 - 3x^(4/3)
a) find critical points
b) find inflection points
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
F%28x%29+=2x%5E2+-+3x%5E%284%2F3%29
a) find critical points
critical points are points where the function is defined and its first derivative is zero or undefined
find F'%28x%29
%28d%2Fdx%29%282x%5E2+-+3x%5E%284%2F3%29%29
=2%28d%2Fdx%29%28x%5E2%29+-+3%28d%2Fdx%29%28x%5E%284%2F3%29%29
=2%2A2x+-+3%284%2F3%29%28x%5E%284%2F3-1%29%29
=4x+-+4%28x%5E%284%2F3-3%2F3%29%29
=4x+-+4%28x%5E%281%2F3%29%29
= 4+%28x+-+x%5E%281%2F3%29%29
F'%28x%29=4+%28x+-+x%5E%281%2F3%29%29
set it equal to zero:
4+%28x+-+x%5E%281%2F3%29%29=0
x+-+root%283%2Cx%29=0
x+=+root%283%2Cx%29..........raise to powe of 3
x%5E3+=x
x%5E3+-x=0
%28x%5E2+-1%29x=0
solutions:
x=0
x%5E2+-1=0=>x%5E2=1=>x=sqrt%281%29=>x=1 or x=-1
so, critical points are :
x=0
x=1
x=-1


b) find inflection points
inflection points are points where the second derivative is zero or undefined
F''%28x%29

%28d%2Fdx%29%284%28x-x%5E%281%2F3%29%29%29

=4%28d%2Fdx%29%28x-x%5E%281%2F3%29%29

=4%28d%2Fdx%29%28x%29+-+4%28d%2Fdx%29%28x%5E%281%2F3%29%29

=4-+4%281%2F3%29%28x%5E%281%2F3-1%29%29

=4-+%284%2F3%29%28x%5E%28-2%2F3%29%29

=4-+%284%2F3%29%281%2Fx%5E%282%2F3%29%29

set it equal to zero
4-+%284%2F3%29%281%2Fx%5E%282%2F3%29%29=0
4=%284%2F3%29%281%2Fx%5E%282%2F3%29%29
4%2F%284%2F3%29=1%2Fx%5E%282%2F3%29
3=1%2Fx%5E%282%2F3%29
root%283%2Cx%5E2%29=1%2F3...........raise to power of 3
x%5E2=1%2F27
x=sqrt%281%2F%283%2A9%29%29
x=1%2F%283sqrt%283%29%29

solutions:
x=1%2F%283sqrt%283%29%29
x=-1%2F%283sqrt%283%29%29