1. ## Problem (Differentiation)

(i) Show that (x - 1) is a factor of x^3 - 4x^2 +4x - 1 and hence find the other factors.

(ii) Hence or otherwise find the maximum and the minimum points of the function f(x) = 2x^(5/2) +5x^2 - 5x +1.

I managed to work out part (i) but I can't see how I can use part(i) to solve part (ii). I am not noticing any relationship between them.

NO IDEA

Thanks in advance for any help.

2. Originally Posted by yobacul
(i) Show that (x - 1) is a factor of x^3 - 4x^2 +4x - 1 and hence find the other factors.

(ii) Hence or otherwise find the maximum and the minimum points of the function f(x) = 2x^(5/2) +5x^2 - 5x +1.

I managed to work out part (i) but I can't see how I can use part(i) to solve part (ii). I am not noticing any relationship between them.
To calculate the extrema of f you need the first derivation first:

$f'(x)=5x^{\frac32} +10x - 5$

Solve the equation f'(x) = 0 for x:

$5x^{\frac32} +10x - 5 = 0$

Use the substitution $u = x^{\frac12}~\implies~x = u^2$ . The equation becomes:

$5u^3 +10u^2 - 5=0~\implies~u^3+2u-1=0$

Now use the same method to solve this equation as you have done with the first one. Obviously one solution is u = -1. Now calculate the other solutions. Re-substitute to get the values of x.