Question

Local max/min of x¹⸍ˣ Use analytical methods to find all local extrema of the function ƒ(x) = x¹⸍ˣ , for x \u003e 0 . Verify your work using a graphing utility.

Accepted Answer

To find the local extrema of the function $$ f(x) = x^{1/x} $$, first take the derivative of the function with respect to $$ x . $$This involves using the chain rule and the power rule. Express $$ f(x) = x^{1/x}  as$$ $$ f(x) = e^{(1/x) \ln(x)}  to $$facilitate differentiation. Differentiate $$ f(x) = e^{(1/x) \ln(x)} $$ using the chain rule: $$ f'(x) = e^{(1/x) \ln(x)} \cdot \left( \frac{d}{dx} \left( \frac{\ln(x)}{x} ight) ight) . $$Calculate $$ \frac{d}{dx} \left( \frac{\ln(x)}{x} ight) $$ using the quotient rule: $$ \frac{d}{dx} \left( \frac{\ln(x)}{x} ight) = \frac{x \cdot \frac{1}{x} - \ln(x) \cdot 1}{x^2} = \frac{1 - \ln(x)}{x^2} . $$Set $$ f'(x) = 0  to $$find critical points: $$ e^{(1/x) \ln(x)} \cdot \frac{1 - \ln(x)}{x^2} = 0 . $$Since $$ e^{(1/x) \ln(x)} 
eq 0 $$, solve $$ \frac{1 - \ln(x)}{x^2} = 0 $$, which simplifies to $$ 1 - \ln(x) = 0 . $$Therefore, $$ \ln(x) = 1 $$, giving $$ x = e . $$Verify the nature of the critical point $$ x = e  by $$using the second derivative test or by analyzing the sign changes of $$ f'(x) $$ around $$ x = e . $$This will confirm whether $$ x = e  is a $$local maximum or minimum. Finally, use a graphing utility to visualize the function and confirm the analytical results.

Local max/min of x¹⸍ˣ Use analytical methods to find all local extrema of the function ƒ(x) = x¹⸍ˣ , for x > 0 . Verify your work using a graphing utility.

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Key Concepts

Critical Points

First Derivative Test

Graphing Utility Verification