Questions: Show that the function f(x)=x^4+6x+4 has exactly one zero in the interval [-1,0]. Which theorem can be used to determine whether a function f(x) has any zeros in a given interval? A. Mean value theorem B. Extreme value theorem C. Rolle's Theorem D. Intermediate value theorem To apply this theorem, evaluate the function f(x)=x^4+6x+4 at each endpoint of the interval [-1,0]. f(-1)= (Simplify your answer.)

Solution Steps

To show that the function \( f(x) = x^4 + 6x + 4 \) has exactly one zero in the interval \([-1, 0]\), we can use the Intermediate Value Theorem. This theorem states that if a continuous function changes sign over an interval, then it has at least one root in that interval. First, evaluate the function at the endpoints of the interval \([-1, 0]\). If the function values at these points have opposite signs, then there is at least one zero in the interval. Additionally, check the derivative to ensure there is no more than one zero.

We evaluate the function \( f(x) = x^4 + 6x + 4 \) at the endpoints of the interval \([-1, 0]\):

\[ f(-1) = (-1)^4 + 6(-1) + 4 = 1 - 6 + 4 = -1 \]

\[ f(0) = 0^4 + 6(0) + 4 = 0 + 0 + 4 = 4 \]

The function values at the endpoints are:

\[ f(-1) = -1 \quad \text{and} \quad f(0) = 4 \]

Since \( f(-1) < 0 \) and \( f(0) > 0 \), the function changes sign over the interval \([-1, 0]\).

By the Intermediate Value Theorem, since \( f(x) \) is continuous and changes sign between \(-1\) and \(0\), there exists at least one \( c \in (-1, 0) \) such that \( f(c) = 0 \).

Next, we check the derivative to ensure there is only one zero in the interval. The derivative is given by:

\[ f'(x) = 4x^3 + 6 \]

Since \( f'(x) \) is a cubic polynomial, we can analyze its behavior. The derivative \( f'(x) \) is always positive for \( x \in [-1, 0] \) because:

\[ f'(-1) = 4(-1)^3 + 6 = -4 + 6 = 2 > 0 \]

This indicates that \( f(x) \) is strictly increasing on the interval \([-1, 0]\).

Final Answer