Determine the position and nature of the stationary points of the function R(x),?

Question

Determine the position and nature of the stationary points of the function R(x),

where

R = x^4 -x^3

Answer 1

murphy's law is better more subjective

Answer 2

you find the stationary points of a function by differentiating and finding where the derivative is zero

if we differentiate R we get:

R'= 4x^3-3x^2

there will be three stationary points (since this is a third order equation) and they occur when:

4x^3-3x^2=0

we can factor out an x^2:

x^2(4x-3)=0

two zeroes (stationary points) occur at x=0

the third is at x=3/4

to find the nature of the stationary points, take the second derivative and eval the second derivative at the stationary points

the second derivative is:

12x^2-6x = 2.25 when x=0.75; when the second deriv is positive, we know this is a MINIMUM

the second derivative is zero when x=0, and the second derivative changes sign as you go from just less than one to just more than one and therefore x=0 is an INFLECTION POINT