Question

given the polynomial f(x)= -3x²+14x+13 what is the smallest positive integer a such that the intermediate value theorem guarantees a zero exists between 1 and a? enter the integer as your andwer. for example, if you found a=8, you woild enter 8.

Answer 1

To find the smallest positive integer such that the Intermediate Value Theorem guarantees a zero exists between 1 and a, you need to determine if the polynomial f(x) = -3x² + 14x + 13 changes sign between these two values. By evaluating f(1) and f(2), you can determine that the smallest positive integer a is 2.

Step-by-step explanation:

To find the smallest positive integer a such that the Intermediate Value Theorem guarantees a zero exists between 1 and a, we need to determine if the polynomial f(x) = -3x² + 14x + 13 changes sign between these two values.

We can do this by evaluating f(1) and f(2). If the sign of f(1) is positive and the sign of f(2) is negative, then there must be at least one zero between 1 and 2.

Substituting x = 1 into the polynomial gives us f(1) = -3(1)² + 14(1) + 13 = 24. Substituting x = 2 gives us f(2) = -3(2)² + 14(2) + 13 = 5. Since f(1) is positive and f(2) is negative, there must be a zero between 1 and 2. Therefore, the smallest positive integer a is 2.