Intermediate value theorem: Let f be a function defined on the closed interval [a, b]. If f is continuous on the closed interval [a, b], then for every real...
Intermediate value theorem:
Let f be a function defined on the closed interval [a, b]. If f is continuous on the closed interval [a, b], then for every real number c between a and b, there exists a number d in (a, b) such that f(d) = c.
Intuitively:
Imagine a continuous function f as a ladder that we can climb up and down. If the ladder starts and ends at the same two points, then the function must have the same value at those points. The Intermediate Value Theorem says that the ladder must stop at some point between the start and end points.
Examples:
If f(x) = x^2 on the interval [0, 1], then f(1/2) = 1/4, and the Intermediate Value Theorem implies that there exists a number d between 0 and 1 such that f(d) = 1/4.
If f(x) = x^3 on the interval [-1, 2], then f(1) = 8, and the Intermediate Value Theorem implies that there exists a number c between -1 and 2 such that f(c) = 8.
If f(x) = x on the interval [0, 1], then f(0) = 0 and f(1) = 1, and the Intermediate Value Theorem implies that there exists a number d between 0 and 1 such that f(d) = 1