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JEE MATHEMATICS MCQ: FUNCTIONS

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Analyzing +4 | -1
Let f(x)=2+cos⁑xf(x)=2+\cos x for all real xx.

Statement -1 : For each real tt, there exists a point c\mathrm{c} in [t,t+Ο€][t, t+\pi] such that fβ€²(c)=0f^{\prime}(c)=0

Statement βˆ’2:f(t)=f(t+2Ο€)-2: f(t)=f(t+2 \pi) for each real tt.
View answer and explanation
Correct option: B
Statement -1 is True, Statement -2 True; Statement - 2 is NOT a correct explanation for Statement - 1
Option B (Statement -1 is True, Statement -2 True; Statement - 2 is NOT a correct explanation for Statement - 1) is correct because both statements are true, but the periodicity of f(x) does not explain why f'(c) = 0 in the given interval.

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If f:(βˆ’βˆž,2]⟢(βˆ’βˆž,4]f:(-\infty, 2] \longrightarrow(-\infty, 4], where f(x)=x(4βˆ’x)f(x)=x(4-x), then fβˆ’1(x)f^{-1}(x) is given by: If g(f(x))=∣sin⁑x∣g(f(x))=|\sin x| and f(g(x))=(sin⁑x)2f(g(x))=(\sin \sqrt{x})^2, then Let f:(2,β€…β€Š3)β†’(0,β€…β€Š1)f:\left( {2,\;3} \right) \to \left( {0,\;1} \right) be defined by f(x)=xβˆ’[x]f\left( x \right) = x - \left[ x \right], then fβˆ’1(x){f^{ - 1}}\left( x \right) equals Let f:Rβ†’Rf: R \rightarrow R be defined as f(x)=x4f(x)=x^4, Choose the correct answer. Which of the following functions are odd functions Let f(x)=(x+1)2βˆ’1(xβ‰₯βˆ’1)f(x)=(x+1)^2-1(x \geq-1). Then the set S={x:f(x)=fβˆ’1(x)}S=\left\{x: f(x)=f^{-1}(x)\right\} contains
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