Which of the following graph represent the function `f (x) = int _(0) ^(sqrtx) e ^((u ^(2))/(x ))` du, for ` x gt 0 and f (0) =o"`

Options

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B.
C.
D.

Answer

To solve the problem, we need to analyze the function given by the integral:

\[ f(x) = \int_0^{\sqrt{x}} e^{\frac{u^2}{x}} \, du \]

for \( x > 0 \) and \( f(0) = 0 \).

### Step 1: Applying the Fundamental Theorem of Calculus

We will differentiate \( f(x) \) with respect to \( x \) using the Leibniz rule (also known as the Fundamental Theorem of Calculus). According to this theorem:

\[ \frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(u, x) \, du \right) = f(b(x), x) \cdot b'(x) - f(a(x), x) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x} \, du \]

In our case, \( a(x) = 0 \), \( b(x) = \sqrt{x} \), and \( f(u, x) = e^{\frac{u^2}{x}} \).

### Step 2: Finding \( f(b(x), x) \) and \( b'(x) \)

1. **Evaluate \( f(b(x), x) \)**:
\[ f(b(x), x) = e^{\frac{(\sqrt{x})^2}{x}} = e^{\frac{x}{x}} = e^1 = e \]

2. **Find \( b'(x) \)**:
\[ b'(x) = \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]

### Step 3: Finding \( f(a(x), x) \) and \( a'(x) \)

1. **Evaluate \( f(a(x), x) \)**:
\[ f(a(x), x) = f(0, x) = e^{\frac{0^2}{x}} = e^0 = 1 \]

2. **Find \( a'(x) \)**:
\[ a'(x) = 0 \]

### Step 4: Finding \( \frac{\partial f}{\partial x} \)

Now we need to find \( \frac{\partial f}{\partial x} \):
\[ \frac{\partial}{\partial x} \left( e^{\frac{u^2}{x}} \right) = e^{\frac{u^2}{x}} \cdot \left(-\frac{u^2}{x^2}\right) \]

### Step 5: Putting it all together

Now we can apply the Leibniz rule:
\[ f'(x) = e \cdot \frac{1}{2\sqrt{x}} - 0 + \int_0^{\sqrt{x}} e^{\frac{u^2}{x}} \left(-\frac{u^2}{x^2}\right) \, du \]

### Step 6: Analyzing the behavior of \( f(x) \)

1. As \( x \to 0 \), \( f(0) = 0 \) is given.
2. For \( x > 0 \), \( f(x) \) is positive and increases as \( x \) increases.

### Step 7: Finding the graph representation

From our analysis, we can see that the function behaves like a parabola. Since \( f(0) = 0 \) and increases for \( x > 0 \), we can represent it as:

\[ y^2 = e^2 x \quad \text{(where \( e^2 \) is a constant)} \]

This is a standard form of a parabola opening to the right.

### Conclusion

The graph of the function \( f(x) \) is a parabola that opens to the right, which corresponds to the equation \( y^2 = 4ax \) where \( a = \frac{e^2}{4} \).