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Question
Using Rolle’s theorem, find the point on the curve y = x(x – 4), x ∈ [0, 4], where the tangent is parallel to x-axis
Solution
We have, y = x(x – 4), x ∈ [0, 4]
Since given function is polynomial it is continuous and differentiable.
Also y(0) = y(4) = 0
So, conditions of Role's theorem are satisfied.
Hence there exists a point c ∈ (0, 4) such that f'(c) = 0
⇒ 2c – 4 = 0
⇒ c = 2
⇒ x = 2 and y(2)
= 2(2 – 4)
= –4
Therefore, the required point on the curve, where the tangent drawn is parallel to the x-axis is (2, – 4).
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