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Question
Solve using intermediate value theorem:
Show that 5x − 6x = 0 has a root in [1, 2]
Solution
Let f(x) = 5x − 6x
5x and 6x are continuous functins for all x ∈ R.
∴ 5x − 6x is also continuous for all x ∈ R.
i.e. f(x) is continuous for all x ∈ R.
A root of f(x) exists if f(x) = 0 for at least one value of x.
f(1) = 51 − 6 (1)
= − 1 < 0
f(2) = (5)2 − 6 (2)
= 13 > 0
∴ f(1) < 0 and f(2) > 0
By intermediate value theorem, there has to be a point ‘c’ between 1 and 2 such that f(c) = 0.
∴ There is a root of the given equation in [1, 2].
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