Question

Let \[ F(x) = \int_0^x e^{t^2 - 3t - 5} \, dt, \quad x>0. \] Then the number of roots of \( F(x) = 0 \) in the interval \( (0, 4) \) is

Hint:

When dealing with integrals of exponential functions, check the behavior of the integrand to determine if the function can have zeros.

Answer 1

Correct Answer: 0

Step 1: Analyze the function.
The function \( F(x) \) is an integral of an exponential function. To determine the number of roots, we need to investigate the behavior of \( F(x) \).
Step 2: Check the behavior of the integrand.
The integrand \( e^{t^2 - 3t - 5} \) is always positive because the exponential function is always positive for any real value of \( t \). Therefore, \( F(x) \) is a strictly increasing function for \( x>0 \).
Step 3: Conclusion.
Since \( F(x) \) is strictly increasing and the integral of a positive function, it will never cross zero. Therefore, there are no roots of \( F(x) = 0 \) in the interval \( (0, 4) \).