# How do you find all critical point and determine the min, max and inflection given #f(x)=x^3+x^2-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]}