how do I find the critical points of f when f(x)= x^4/5(x-5)^2?

Question

I did the product rule and got
4/5x^-1/5(x-5)^2 + x^(4/5) (2x-10)
The answer is supposed to be x=0, 10/7, 5
Plz show me clearly what to do next to get the crit pt. Thnx

Answer 1

ok well, in this case you can't use the product rule (or you can but you have to use negative exponents and its easier of you use the quotient rule).

so you should use the quotient rule.

(bottom * derivative of top - top * derivative of bottom) divided by bottom squared.

so in this case, I will call the numerator g(x) and the denominator h(x).

[g'(x)*h(x) - h'(x) * g(x)] / [h(x)]^2

so g(x) is x^4, g'(x) = 4x^3.
h(x) = 5(x-5)^2, h'(x) = 10(x-5)

so f '(x) = [(4x^3)(5(x-5)^2) - (x^4)(10(x-5))] / [5(x-5)^2]^2
= [(4x^3)(5(x-5)^2) - (x^4)(10(x-5))] / [25(x-5)^4]

you can cancel out a x-5 term.
= [(4x^3)(5(x-5)) - (x^4)(10)] / [25(x-5)^3]
= [20x^4 - 100 x^3 - 10x^4] / [25(x-5)^3]
= [5(4x^4 - 20 x^3 - 2x^4)] / [(5)(5)(x-5)^3]
= [2x^4 - 20x^3] / [5(x-5)^3]
= [2x^3(x-10)] / [5(x-5)^3]

So this is the factored form.

so to find the critical point, you have to find the roots of the top and bottom. the roots of the top are x=0 and x=10. and the root of the bottom is x=5.

Hope that helped. I have no idea why you got x=10/7, but the method is still right. maybe I typod somewhere or you typod in the equation or somethin. sorry

Answer 2

You're off to a good start; you have the correct derivative.

The critical points will occur when f is not differentiable or when f'(x) = 0.
f'(x) = 4/5x^-1/5(x-5)^2 + x^(4/5) (2x-10)
Here's a trick for fractional powers: When you have fractional powers separated by integer amounts (as here: 4/5 and -1/5, very common in differentiation for the same reason you have it here), take out a factor of the lowest power from all related terms. Here this is x^(-1/5); we think of x^(4/5) as x^(-1/5).x.

f'(x) = 4/5x^-1/5(x-5)^2 + x^(4/5) (2x-10)
= x^(-1/5) . (4/5 (x-5)^2 + x(2x-10))
= x^(-1/5) . (x-5) (4/5(x-5) + 2x)
= x^(-1/5) . (x-5) . (14/5 x - 4)
So critical points are at x = 0, x = 5 and x = 4/(14/5) = 20/14 = 10/7.

Answer 3

to find the critical points take the derivative of the equation the find the roots (where the equation is equal to zero). so instead of mulitplying it out try it this way.

x^4/5* 2(x-5) + (x-5)^2*4/5(x)^-1/5

and the critical points should be x = 0 and x = 5