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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Using Rolle’s theorem, find the point on the curve $\text{y}=\text{x}(\text{x}-4),\text{x}\in[0,4].$ where the tangent is parallel to x-axis.
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Answer

We have, $\text{y}=\text{x}(\text{x}-4),\text{x}\in[0,4]$
Since given function is polynomial it is continuous and differentiable
Also y(0) = y(4) = 0
So, conditions of Rolle's theorem are satisfied.
Hence there exists a point $\text{c}\in(0,4)$ 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-axix is (2, -4).

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