Find the Critical Points 3x^(2/3)-2x

Problem

3*x(2/3)−2*x

Solution

1. Identify the function ƒ(x)=3*x(2/3)−2*x and the definition of critical points, which occur where the derivative ƒ(x)′=0 or where ƒ(x)′ is undefined.

2. Differentiate the function using the power rule d(xn)/d(x)=n*x(n−1)

ƒ(x)′=3⋅2/3*x(−1/3)−2

3. Simplify the derivative expression.

ƒ(x)′=2*x(−1/3)−2

4. Rewrite the derivative using a radical to identify where it is undefined.

ƒ(x)′=2/√(3,x)−2

5. Determine where the derivative is undefined. The expression 2/√(3,x)−2 is undefined when the denominator is zero, which occurs at x=0

6. Solve for x where the derivative equals zero.

2/√(3,x)−2=0

7. Isolate the radical term.

2/√(3,x)=2

8. Simplify the equation by dividing both sides by 2.

1/√(3,x)=1

9. Solve for x by cubing both sides.

x=1

10. List all values of x in the domain of ƒ(x) where ƒ(x)′=0 or ƒ(x)′ is undefined.

Final Answer

Critical Points: *x=0,x=1

Want more problems? Check here!