Limits & Continuity · Continuity & Theorems
Intermediate Value Theorem
If f is continuous on [a,b] and N is between f(a) and f(b), then there exists at least one c in (a,b) where f(c) = N. Often used to show a root exists.
Conditions. f must be continuous on the closed interval [a, b].
- 0 variables
- 2 worked examples
- 18 in Limits & Continuity
Worked examples
Show that x³ + x - 1 = 0 has a root in (0, 1).
- Let f(x) = x³ + x - 1. f is a polynomial, so continuous everywhere.
- f(0) = 0 + 0 - 1 = -1 < 0
- f(1) = 1 + 1 - 1 = 1 > 0
- Since f(0) < 0 < f(1) and f is continuous on [0,1], by IVT there exists c in (0,1) with f(c) = 0
Answer: By IVT, there is at least one root in (0, 1).
1 more worked examplePremium
Show that cos(x) = x has a solution in (0, π/2).
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