Find the Local Maxima and Minima f(x) = cube root of x-1

Problem

ƒ(x)=√(3,x−1)

Solution

1. Rewrite the function using a rational exponent to make differentiation easier.

ƒ(x)=(x−1)(1/3)

2. Find the derivative of the function using the power rule and the chain rule.

d(ƒ(x))/d(x)=1/3*(x−1)(−2/3)

3. Simplify the derivative expression by moving the negative exponent to the denominator.

ƒ(x)′=1/(3*(x−1)(2/3))

4. Identify critical points by finding where the derivative is zero or undefined.

3*(x−1)(2/3)=0⇒x=1

5. Analyze the derivative to determine the behavior of the function. The derivative ƒ(x)′ is always positive for all x≠1 because the term (x−1)(2/3) is a square of a cube root, which is always non-negative.

ƒ(x)′>0* for *x<1

ƒ(x)′>0* for *x>1

6. Conclude that since the derivative does not change sign at the critical point x=1 and the function is strictly increasing, there are no local extrema.

Final Answer

Local Maxima: None, Local Minima: None

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