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\(f:\mathbb{R}\rightarrow\mathbb{R}\) is increasing in \((-\infty,0)\cup(1,\infty)\) and decreasing in \((0,1)\). If \(f(2)=6\) and the continuous curve \(y=f(x)\) passes through \((0,0)\), then

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Whenever values of a function at two points are given, immediately think of the Mean Value Theorem: \[ f'(c)=\frac{f(b)-f(a)}{b-a}. \] This theorem is frequently used in objective-type calculus questions.

Updated On: Jun 25, 2026

A point \(a\in(0,2)\) such that \(f'(a)=3\)
A point \(a\in(0,2)\) such that \(f'(a)=0\)
\(f'(0)<0\)
\(f'(1)>0\)

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The Correct Option is A

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\(f:\mathbb{R}\rightarrow\mathbb{R}\) is increasing in \((-\infty,0)\cup(1,\infty)\) and decreasing in \((0,1)\). If \(f(2)=6\) and the continuous curve \(y=f(x)\) passes through \((0,0)\), then

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The Correct Option is A

Solution and Explanation

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