Final answer:
According to the Intermediate Value Theorem, since f(x) = 5x^3 + 3x^2 + 5x + 4 changes sign from f(-1) = -3 (negative) to f(0) = 4 (positive), there must be at least one real zero between -1 and 0.
Step-by-step explanation:
To show that the function f(x) = 5x^3 + 3x^2 + 5x + 4 has a real zero between -1 and 0 using the Intermediate Value Theorem (IVT), we evaluate the function at the endpoints of the interval [-1, 0].
First, let's evaluate f at x = -1:
f(-1) = 5(-1)^3 + 3(-1)^2 + 5(-1) + 4
= -5 + 3 - 5 + 4
= -3
Now, let's evaluate f at x = 0:
f(0) = 5(0)^3 + 3(0)^2 + 5(0) + 4
= 0 + 0 + 0 + 4
= 4
According to the IVT, if a continuous function changes sign over an interval, there must be at least one root in that interval. In this case, f(-1) = -3 and f(0) = 4, which shows a change in sign from negative to positive. Thus, there must be at least one real zero between -1 and 0.