ધારો કે \(\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) અચળ ન હોય તેવો દ્વિ વિકલનીય વિધેય છે જ્યાં \(\mathrm{g}\left(\frac{1}{2}\right)=\mathrm{g}\left(\frac{3}{2}\right)\). જો વાસ્તવિક મૂલ્યવાળું વિધેય \(F\) એ \(f(x)=\frac{1}{2}[g(x)+\mathrm{g}(2-x)]\) ] પ્રમાણે વ્યાખ્યાયિત થાય, તો :

A \((0,2)\) માં ઓછામાં ઓછા બે \(x\) માટે \(f^{\prime \prime}(x)=0\)
B \((0,1)\) માં બરાબર એક જ \(x\) માટે \(f^{\prime \prime}(x)=0\)
C \((0,1)\) માં શૂન્ય નથી. કોઈ પણ \(x\) માટે \(f^{\prime \prime}(x)=0\)
D \(f^{\prime}\left(\frac{3}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)=1\)

Verified Solution

Answer & Solution

Correct Answer

(A) \((0,2)\) માં ઓછામાં ઓછા બે \(x\) માટે \(f^{\prime \prime}(x)=0\)

Step-by-step Solution

Detailed explanation

\({ }^{\prime}(x)=\frac{g^{\prime}(x)-g^{\prime}(2-x)}{2}, f^{\prime}\left(\frac{3}{2}\right)=\frac{g^{\prime}\left(\frac{3}{2}\right)-g^{\prime}\left(\frac{1}{2}\right)}{2}=0\) Also…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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