If `f(x)=x^3+7x-1,` then `f(x)` has a zero between `x=0a n dx=1` . The theorem that best describes this is (a) mean value theorem (b) maximum-minimum value theorem (c) intermediate value theorem (d) none of these

Options

A. mena value theorem
B. maximum-minimum value theorem
C. intermediate value theorem
D. none of these

Answer

To solve the problem, we need to determine which theorem best describes the situation where the function \( f(x) = x^3 + 7x - 1 \) has a zero between \( x = 0 \) and \( x = 1 \).

### Step 1: Evaluate the function at the endpoints
First, we will evaluate \( f(x) \) at the endpoints \( x = 0 \) and \( x = 1 \).

- Calculate \( f(0) \):
\[ f(0) = 0^3 + 7 \cdot 0 - 1 = -1 \]

- Calculate \( f(1) \):
\[ f(1) = 1^3 + 7 \cdot 1 - 1 = 1 + 7 - 1 = 7 \]

### Step 2: Analyze the values obtained
Now we have:
- \( f(0) = -1 \) (which is less than 0)
- \( f(1) = 7 \) (which is greater than 0)

### Step 3: Apply the Intermediate Value Theorem
Since \( f(0) < 0 \) and \( f(1) > 0 \), by the Intermediate Value Theorem (IVT), we can conclude that there exists at least one \( c \) in the interval \( (0, 1) \) such that \( f(c) = 0 \).

### Conclusion
The theorem that best describes this situation is the **Intermediate Value Theorem**.

Thus, the correct answer is:
(c) Intermediate Value Theorem