Let `f(x)` be a non-positive continuous function and `F(x)=int_(0)^(x)f(t)dt AA x ge0` and `f(x) ge cF(x)` where `c lt 0` and let `g:[0, infty) to R` be a function such that `(dg(x))/(dx) lt g(x) AA x gt 0` and `g(0)=0`
The total number of root(s) of the equation `f(x)=g(x)` is/ are

Options

A. `infty`
B. 1
C. 2
D. 0

Explanation

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) based on the given conditions and find the total number of roots of the equation \( f(x) = g(x) \).

### Step-by-Step Solution:

1. **Understanding the Function \( F(x) \)**:
- Given \( F(x) = \int_0^x f(t) dt \) where \( f(x) \) is a non-positive continuous function. This implies \( F(x) \) is non-increasing since \( f(t) \leq 0 \).

2. **Behavior of \( F(x) \)**:
- Since \( f(x) \leq 0 \), the integral \( F(x) \) is non-increasing. Hence, \( F(x) \) is either constant or decreasing. Moreover, since \( f(x) \) is continuous and non-positive, \( F(0) = 0 \) (because \( F(0) = \int_0^0 f(t) dt = 0 \)).

3. **Analyzing the Inequality \( f(x) \geq cF(x) \)**:
- We know \( c < 0 \). This implies \( cF(x) \) is non-positive (since \( F(x) \geq 0 \)). Therefore, \( f(x) \) is bounded below by a non-positive value, which reinforces that \( f(x) \) is non-positive.

4. **Behavior of \( g(x) \)**:
- The function \( g(x) \) satisfies \( \frac{dg(x)}{dx} < g(x) \) for \( x > 0 \) and \( g(0) = 0 \). This indicates that \( g(x) \) is growing slower than the exponential function \( e^x \) because if \( g(x) \) were to grow faster, its derivative would be greater than \( g(x) \).

5. **Finding the Roots of \( f(x) = g(x) \)**:
- Since \( f(x) \) is non-positive and \( g(0) = 0 \), we can conclude that \( g(x) \) starts at 0 and increases. However, due to the condition \( \frac{dg(x)}{dx} < g(x) \), \( g(x) \) will grow but at a rate slower than an exponential function.
- As \( x \) increases, \( g(x) \) will eventually become positive, but since \( f(x) \) is non-positive, the two functions can intersect at most once.

6. **Conclusion**:
- The total number of roots of the equation \( f(x) = g(x) \) is **1**. This is because \( f(x) \) is non-positive and \( g(x) \) starts from zero and increases, leading to exactly one intersection point.

### Final Answer:
The total number of root(s) of the equation \( f(x) = g(x) \) is **1**.