SOLUTION: Plot a graph of the function f(x) =2x^2 - 3x^4/3 and identify the location of critical points and inflections points.

Algebra ->  Probability-and-statistics -> SOLUTION: Plot a graph of the function f(x) =2x^2 - 3x^4/3 and identify the location of critical points and inflections points.       Log On


   

Question 1180890: Plot a graph of the function f(x) =2x^2 - 3x^4/3 and identify the location of critical points and inflections points.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Plot a graph of the function f%28x%29+=%282x%5E2+-+3x%5E4%29%2F3 and identify the location of critical points and inflections points.

+graph%28+600%2C+600%2C+-3%2C+3%2C+-3%2C+3%2C+%282x%5E2+-+3x%5E4%29%2F3%29+

Solving the equation f′%28c%29=0 on this interval, we get the location of critical points.
f%28x%29+=%282x%5E2+-+3x%5E4%29%2F3
f%28x%29+=%282%2F3%29x%5E2+-+x%5E4
f+' =2%282%2F3%29x+-+4x%5E3
f ' =%284%2F3%29x+-+4x%5E3

%284%2F3%29x+-+4x%5E3=0
x%284%2F3+-+4x%5E2%29=0
one solution is x=0
%284%2F3+-+4x%5E2%29=0=>4%2F3+=4x%5E2=>4%2F%284%2A3%29+=x%5E2=>1%2F3+=x%5E2

solutions:
x=sqrt%281%2F3%29
x = ± 1%2Fsqrt%283%29

the location of critical points is at x+=+0,x=1%2Fsqrt%283%29, and -1%2Fsqrt%283%29

An inflection point is a point on the graph of a function at which the concavity changes. Points of inflection can occur where the second derivative is zero. In other words, solve f '' =+0 to find the potential inflection points.
Even if f ''%28c%29+=+0, you can't conclude that there is an inflection at x+=+c.

f '' =1%2A%284%2F3%29+-+3%2A4x%5E2+
f+'' =4%2F3+-+12x%5E2+
4%2F3+-+12x%5E2+=0
4%2F3+=12x%5E2
x%5E2=%284%2F3%29+%2F12
x%5E2=4%2F%283%2A12%29
x%5E2=1%2F%283%2A3%29
x%5E2=1%2F9
x = ± 1%2F3
there is an inflection at x+=+1%2F3 and at x+=+-1%2F3