Find the point on the curve y = cos x – 1, x $$\in$$ $$[{\pi\over 2}, {3\pi\over 2}]$$ at which tangent is parallel to the x-axis.

Solution :

Let f(x) = cos x – 1, Clearly f(x) is continous on $$[{\pi\over 2}, {3\pi\over 2}]$$ and differentiable on $$({\pi\over 2}, {3\pi\over 2})$$.

Also, f$$(\pi\over 2)$$ = $$cos {\pi\over 2}$$ – 1 = -1 = f$$(3\pi\over 2)$$.

Thus, all the conditions of rolle’s theorem are satisfied. Consequently,there exist at least one point c $$\in$$ $$({\pi\over 2}, {3\pi\over 2})$$ for which f'(c) = 0. But,

f'(c) = 0 $$\implies$$  -sin c = 0  $$\implies$$  c = $$\pi$$

$$\therefore$$   f(c) = $$cos \pi$$ – 1 = -2

By the geometric interpretation of rolle’s theorem ($$\pi$$, -2) is the point on y = cos x – 1 where tangent is parallel to x-axis.

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