If `f(x)= int_(0)^(x)(f(t))^(2) dt, f:R rarr R ` be differentiable function and `f(g(x))` is differentiable at `x=a`, then

Options

A. (a)g(x) must be differentiable at x = a
B. (b)g(x) is discontinuous, then f(a) = 0
C. (c)`f(a) ne 0`, then g(x) must be differentiable
D. (d)None of these

Answer

To solve the problem, we start with the given function:

\[ f(x) = \int_0^x (f(t))^2 \, dt \]

### Step 1: Differentiate \( f(x) \)

Using the Fundamental Theorem of Calculus, we differentiate \( f(x) \):

\[ f'(x) = (f(x))^2 \]

### Step 2: Analyze the differentiability of \( f(g(x)) \)

We are given that \( f(g(x)) \) is differentiable at \( x = a \). By applying the chain rule, we have:

\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]

### Step 3: Substitute \( f'(g(x)) \)

From Step 1, we know that:

\[ f'(g(x)) = (f(g(x)))^2 \]

Thus, we can rewrite the derivative as:

\[ \frac{d}{dx} f(g(x)) = (f(g(x)))^2 \cdot g'(x) \]

### Step 4: Evaluate at \( x = a \)

Since \( f(g(x)) \) is differentiable at \( x = a \), we can evaluate the expression at \( x = a \):

\[ \frac{d}{dx} f(g(x)) \bigg|_{x=a} = (f(g(a)))^2 \cdot g'(a) \]

### Step 5: Conclusion about differentiability

For \( f(g(x)) \) to be differentiable at \( x = a \), both \( f(g(a)) \) and \( g'(a) \) must exist. This implies that \( g(x) \) must be differentiable at \( x = a \).

### Final Answer

Thus, we conclude that \( g(x) \) must be differentiable at \( x = a \).

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