Posted by **dana** on Saturday, December 18, 2010 at 3:42am.

(a) Find the critical points of f(x) = x^(1/3)-x^(1/5)

(b) Determine the maximum and minimum values of f on the interval [0; 2] and where they occur.

- calculus -
**Reiny**, Saturday, December 18, 2010 at 9:04am
I will assume that you know that critical points would be points such as the x and y intercepts, the max and min points, and any points of inflection.

I think the problem here lies in trying to solve different equations containing fractional exponents.

the y-intercept is easy, let x = 0

y = 0

for the x-intercept let y = 0

so x^(1/2 - x^(1/5) = 0

x^(1/3) = x^(1/5)

raise it to the 15th power, (15 is the LCM of 3 and 5)

x^5 = x^3

x^5 - x^3 - 0

x^3(x^2 - 1) = 0

x = 0 or x = ± 1

After you get dy/dx = (1/3)x^(-2/3) - (1/5)x^(-4/5)

you will have to set this equal to zero as well

so the same sort of thing as above

x^(-2/3) = x^(-4/5) , after dividing out the 1/3

so 1/x^(2/3) = 1/x^(4/5)

x^(4/5) = x^(2/3) after cross-multiplying

raise to the 15th

x^12 = x^10

x^10(x^2 - 1) = 0

x = 0 or x = ± 1 (well, isn't that interesting)

etc.

## Answer this Question

## Related Questions

- Please check my Calculus - 1. Find all points of inflection: f(x)=1/12x^4-2x^2+...
- Please check my Calculus - 1. Find all points of inflection of the function f(x...
- calculus - there are no examples of this type of problem in my book so if you ...
- Calculus (pleas help!!!) - Find the absolute maximum and absolute minimum ...
- Calculus (pleas help!!!) - Find the absolute maximum and absolute minimum values...
- Calculus - Find the absolute maximum and minimum values of f on the given closed...
- Calculus - Given the function f(x)= x√(x+3), Indicate the domain of f(x...
- Relative Extrema: AP Calculus - Consider the function f(x)= (3/4)x^4-x^3-3x^2+6x...
- Calculus - Find the absolute maximum and absolute minimum values of the function...
- Calculus - Find any absolute maximum and minimum and local maximum and minimum ...