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Question
Show that there is a root for the equation x3 − 3x = 0 between 1 and 2.
Solution
Let f(x) = x3 − 3x
f(x) is a polynomial function and hence it 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) = (1)3 – 3(1)
= – 2 < 0
f(2) = (2)3 – 3(2)
= 2 > 0
∴ f(1) < 0 and f(2) > 0
∴ By intermediate value theorem, there has to be point ‘c’ between 1 and 2
Such that f(c) = 0
∴ There is a root of the given equation between 1 and 2.
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