Maximum and minimum as ordered pairs

Question

y= x⁴ -4x³

Answer 1

Abi, simply take the derivative of y with respect to x, and set it to equal to zero to find the values of x's whenever the original function reaches it maximum and minimum values.

 

dy/dx=4x³-12x²= 0 This implies 4x²(x-3)=0 Thus, the extreme values occur at x=0 and x=3

 

at x=0, y=0 (0,0) Note: This may be an inflection point, not a local maximum or minimum point

 

at x=3, y=3⁴-4(3)³= 3³(3-4)=27(-1)=-27 (3,-27)

 

To determine whether point (0,0) is whether it is a maximum, minumim, or inflection point, is to evaluate the function on both sides of x=0. Evaluate y at x=-0.1 in which is y=(-0.1)⁴-4(-0.1)³=4.1*10^-4 and at x=(0.1) in which y=(0.1)⁴-3*(0.1)³=-3.9*10^-3 Thus, point(0,0) is an inflection point.

 

Let x=2.9 in which y=2.9⁴-4*(2.9)³=-26828 and at x=3.1 y=-26.812. This implies point (3,-27) is a local minimum.