Questions: Find the local minimum and maximum values of f. f(x)=x^top ln(x)

Transcript text: Find the local minimum and maximum values of $f$. \[ f(x)=x^{\top} \ln (x) \]

Solution

Solution Steps

To find the local minimum and maximum values of the function $ f(x) = x^{\top} \ln(x) $, we need to follow these steps:

Compute the first derivative of the function $ f(x) $.
Set the first derivative equal to zero to find the critical points.
Compute the second derivative of the function $ f(x) $.
Use the second derivative test to determine whether each critical point is a local minimum, local maximum, or neither.

Step 1: Define the Function

We start with the function defined as: \[ f(x) = x \ln(x) \]

Step 2: Compute the First Derivative

The first derivative of the function is calculated as: \[ f'(x) = \ln(x) + 1 \]

Step 3: Find Critical Points

To find the critical points, we set the first derivative equal to zero: \[ \ln(x) + 1 = 0 \implies \ln(x) = -1 \implies x = e^{-1} \] Thus, the critical point is: \[ x = e^{-1} \]

Step 4: Compute the Second Derivative

The second derivative of the function is: \[ f''(x) = \frac{1}{x} \]

Step 5: Determine the Nature of the Critical Point

We evaluate the second derivative at the critical point $ x = e^{-1} $: \[ f''(e^{-1}) = \frac{1}{e^{-1}} = e \] Since $ f''(e^{-1}) > 0 $, this indicates that $ x = e^{-1} $ is a local minimum.

Final Answer

The local minimum occurs at: \[ \boxed{x = e^{-1}} \]

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