x x \sqrt [ x ]{ x }

Calculus Level 2

If f ( x ) = x x f\left( x \right) =\sqrt [ x ]{ x } , at what value of x x (where x > 1 x>1 ) would f ′ ( x ) f'\left( x \right) be equal to 0? Round to 3 decimal places.


The answer is 2.718.

Louis Ullman by Louis Ullman , 16, USA

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1 solution

Vincent Moroney
Jun 25, 2018

f ( x ) = x 1 x f ( x ) = e 1 x ln ⁡ ( x ) f ′ ( x ) = e 1 x ln ⁡ ( x ) [ 1 x 2 ( 1 − ln ⁡ ( x ) ) ] f ′ ( x ) = x 1 x ( 1 x 2 ( 1 − ln ⁡ ( x ) ) ) f ′ ( x ) = 0 ⟺ 1 − ln ⁡ ( x ) = 0 ⇒ x = e \begin{aligned} f(x) = & x^{\frac{1}{x}} \\ f(x) = & e^{\frac{1}{x}\ln(x)} \\ f'(x) = & e^{\frac{1}{x}\ln(x)} \Big[ \frac{1}{x^2}\Big( 1-\ln(x)\Big)\Big]\\ f'(x) = & x^{\frac{1}{x}}\Big(\frac{1}{x^2}\big(1-\ln(x)\big)\Big) \\ f'(x) = & 0 \iff 1-\ln(x) = 0 \Rightarrow \boxed{x = e} \end{aligned}

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