QuestionFind the maximum value of $y = x^{1/x}$ for $x > 0$ .

Studdy Solution

STEP 1

Assumptions1. The function is $y=x^{1 / x}$ , where $x >0$ . We are looking for the maximum value of this function

STEP 2

To find the maximum value of a function, we can use calculus. Specifically, we can take the derivative of the function, set it equal to zero, and solve for $x$ . This will give us the critical points of the function.
First, let's rewrite the function in a form that's easier to differentiate. We can use the property that $a^{b}=e^{b \ln a}$ to rewrite the function as follows $y = e^{\frac{\ln x}{x}}$

STEP 3

Now, we can take the derivative of $y$ with respect to $x$ . We will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
The derivative of $e^{u}$ is $e^{u} \cdot u'$ , where $u'$ is the derivative of $u$ .
Let $u = \frac{\ln x}{x}$ . Then $u' = \frac{1 - \ln x}{x^{2}}$ .
So, the derivative of $y$ is $y' = e^{\frac{\ln x}{x}} \cdot \frac{1 - \ln x}{x^{2}}$

STEP 4

To find the critical points, we set the derivative equal to zero and solve for $x$ $e^{\frac{\ln x}{x}} \cdot \frac{1 - \ln x}{x^{2}} =0$

STEP 5

The exponential function $e^{u}$ is never zero, so we can divide both sides by $e^{\frac{\ln x}{x}}$ to simplify the equation $\frac{1 - \ln x}{x^{2}} =0$

STEP 6

olving the above equation, we get $1 - \ln x =0$

STEP 7

olving for $x$ , we get $x = e$

STEP 8

We need to check if $x = e$ is a maximum, minimum, or neither. We can do this by taking the second derivative of $y$ and evaluating it at $x = e$ .
The second derivative of $y$ is $y'' = e^{\frac{\ln x}{x}} \cdot \left( \frac{2 \ln x -3}{x^{3}} \right)$

STEP 9

valuating the second derivative at $x = e$ , we get $y''(e) = e^{\frac{\ln e}{e}} \cdot \left( \frac{2 \ln e -3}{e^{3}} \right) = -\frac{}{e^{2}} <$

STEP 10

Since the second derivative at $x = e$ is negative, $x = e$ is a local maximum. As the function $y = x^{/x}$ is decreasing for $x > e$ , we can conclude that the maximum value of $y$ is at $x = e$ .

STEP 11

Substitute $x = e$ into the original function to find the maximum value $y_{max} = e^{/e}$ The maximum value of $y = x^{/x}$ for $x >0$ is $e^{/e}$ .

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