Question

what is the smallest positive integer a such that the intermediate value theorem guarantees a zero exists between 0 and a?

Answer 1

The smallest positive integer that the intermediate value theorem guarantees a zero exists between 0 and a is 3.

What is the intermediate value theorem?

Intermediate value theorem is theorem about all possible y-value in between two known y-value.

x-intercepts

-x^2 + x + 2 = 0

x^2 - x - 2 = 0

(x + 1)(x - 2) = 0

x = -1, x = 2

y intercepts

f(0) = -x^2 + x + 2

f(0) = -0^2 + 0 + 2

f(0) = 2

(Graph attached)

From the graph we know the smallest positive integer value that the intermediate value theorem guarantees a zero exists between 0 and a is 3

For proof, the zero exists when x = 2 and f(3) = -4 < 0 and f(0) = 2 > 0.

Your question is not complete, but most probably your full questions was

Given the polynomial f(x)=− x 2 +x+2 , what is the smallest positive integer a such that the Intermediate Value Theorem guarantees a zero exists between 0 and a ?

Thus, the smallest positive integer that the intermediate value theorem guarantees a zero exists between 0 and a is 3.

Learn more about intermediate value theorem here:

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