Let `f(x),xgeq0,` be a non-negative continuous function, and let `f(x)=int_0^xf(t)dt ,xgeq0,` if for some `c >0,f(x)lt=cF(x)` for all `xgeq0,` then show that `f(x)=0` for all `xgeq0.`

Explanation

To solve the problem, we need to show that if \( f(x) \) is a non-negative continuous function defined by the integral \( f(x) = \int_0^x f(t) \, dt \) and satisfies the condition \( f(x) < cF(x) \) for some \( c > 0 \) and all \( x \geq 0 \), then \( f(x) = 0 \) for all \( x \geq 0 \).

### Step-by-Step Solution:

1. **Understanding the given function**:
We are given that \( f(x) = \int_0^x f(t) \, dt \). This implies that \( f(x) \) is the area under the curve of \( f(t) \) from \( 0 \) to \( x \).

2. **Differentiating both sides**:
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