Answer to Question #103920 in Calculus for Martin

Question

Find and classify the critical points off(x)=9x4(2−x)5 as local maxima and minima.
Critical points: x

Answer 1

The critical points at f'(x)=0,

$(9*x^4)'*(2-x)^5+9*x^4*((2-x)^5)'=0, 36*x^3*(2-x)^5-45*x^4*(2-x)^4=0, 9*x^3*(2-x)^4*(8-9*x)=0,$

$f'(-1)<0; f'(0,5)>0; f'(1)<0; f'(3)<0.$

When passing through point 0, the derivative changes the sign from - to +, then

x₁=0 is a point of minimum.

When passing through point 8/9, the derivative changes the sign from + to -, then

x₂=8/9 is a point of maximum.

When passing through point 0, the derivative does not change the sign, then

x₃=2 is not a point of minimum/maximum.