Find Where Increasing/Decreasing Using Derivatives f(x)=x^(1/3)(x-4)

Problem

ƒ(x)=x^(1/3)*(x−4)

Solution

1. Rewrite the function by distributing x^(1/3) to make differentiation easier.

ƒ(x)=x^(4/3)−4*x^(1/3)

2. Differentiate the function using the power rule to find the first derivative.

d(ƒ(x))/d(x)=4/3*x^(1/3)−4/3*x^(−2/3)

3. Factor the derivative to identify critical points.

ƒ(x)^′=4/3*x^(−2/3)*(x−1)

4. Simplify the expression into a single fraction.

ƒ(x)^′=(4*(x−1))/(3*x^(2/3))

5. Identify the critical points where ƒ(x)^′=0 or ƒ(x)^′ is undefined.

x−1=0⇒x=1

3*x^(2/3)=0⇒x=0

6. Test the intervals (−∞,0) (0,1) and (1,∞) by checking the sign of ƒ(x)^′
   For x=−1 ƒ^′*(−1)=(4*(−2))/(3*(1))<0 (Decreasing)
   For x=0.5 ƒ(0.5)^′=(4*(−0.5))/(3*(0.5)^(2/3))<0 (Decreasing)
   For x=2 ƒ(2)^′=(4*(1))/(3*(2)^(2/3))>0 (Increasing)

7. Conclude the intervals based on the sign of the derivative.

Final Answer

Increasing: *(1,∞), Decreasing: *(−∞,0)∪(0,1)

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