How do you find all critical point and determine the min, max and inflection given f(x)=x^3+x^2-xf(x)=x3+x2−x?
2 Answers
Critical Points are:
Explanation:
We have
To identify the critical vales, we differentiate and find find values of
Differentiating wrt
f'(x) = 3x^2 + 2x - 1 .... [1]
At a critical point,
f'(x)=0 => 3x^2 + 2x - 1 = 0
:. (3x-1)(x+1) = 0
x=-1,1/3
Ton find the y-coordinate we substitute the required value into
So the critical points are
We identify the nature of these critical points by looking at the sign of second derivative, and
Differentiating [1] wrt
f''(x) = 6x + 2
x=-1 => f''(-2)=-6+2 < 0 , ie a maximum
x=1/3 => f''(1)= 2+2>0 , ie a minimum
Incidental, As this is a cubic with a positive coefficient of
The max is at
The min is at
The inflexion point is at
Explanation:
We have to calculate the first and second derivative.
So, we do a sign chart
So, we have a max at
To determine the inflexion points, we calculate
The inflexion point is at
Also,
and
graph{x^3+x^2-x [-2.43, 2.436, -1.217, 1.215]}