The minimum value of `sqrt(e^(x^(2)) -1)` is

Answer

To find the minimum value of the function \( f(x) = \sqrt{e^{x^2} - 1} \), we will follow these steps:

### Step 1: Define the function
We start by defining the function:
\[ f(x) = \sqrt{e^{x^2} - 1} \]

### Step 2: Differentiate the function
Next, we need to find the derivative of \( f(x) \). We will use the chain rule for differentiation. The derivative of \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx} \), where \( u = e^{x^2} - 1 \).

First, we differentiate \( u \):
\[ u = e^{x^2} - 1 \implies \frac{du}{dx} = e^{x^2} \cdot 2x = 2x e^{x^2} \]

Now, applying the chain rule:
\[ f'(x) = \frac{1}{2\sqrt{e^{x^2} - 1}} \cdot 2x e^{x^2} \]
This simplifies to:
\[ f'(x) = \frac{x e^{x^2}}{\sqrt{e^{x^2} - 1}} \]

### Step 3: Find critical points
To find the critical points, we set the derivative equal to zero:
\[ \frac{x e^{x^2}}{\sqrt{e^{x^2} - 1}} = 0 \]
This implies:
\[ x e^{x^2} = 0 \]
Since \( e^{x^2} \) is never zero, we have:
\[ x = 0 \]

### Step 4: Evaluate the function at the critical point
Now we evaluate \( f(x) \) at the critical point \( x = 0 \):
\[ f(0) = \sqrt{e^{0^2} - 1} = \sqrt{e^0 - 1} = \sqrt{1 - 1} = \sqrt{0} = 0 \]

### Step 5: Conclusion
Thus, the minimum value of \( f(x) = \sqrt{e^{x^2} - 1} \) is:
\[ \boxed{0} \]

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