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Differentiation question

2024 · 30 Jan · Shift 1 · Q31

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JEE Main
Mathematics
Differentiation
2024 · 30 Jan · Shift 1 · Q31

Differentiation question

2024 · 30 Jan · Shift 1 · Q31

JEE MainMathematicsDifferentiationMCQ+4 / −1

Let

g: \mathbf{R} \rightarrow \mathbf{R}

be a non constant twice differentiable function such that

\mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\mathrm{g}^{\prime}\left(\frac{3}{2}\right)

. If a real valued function

f

is defined as

f(x)=\frac{1}{2}[g(x)+g(2-x)]

, then

A
$f^{\prime \prime}(x)=0$ for atleast two $x$ in $(0,2)$
B
$f^{\prime}\left(\frac{3}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)=1$
C
$f^{\prime \prime}(x)=0$ for no $x$ in $(0,1)$
D
$f^{\prime \prime}(x)=0$ for exactly one $x$ in $(0,1)$

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