In organic chemistry, functional groups can influence the reactivity of a benzene ring as ortho/para or meta directors based on their electron-withdrawing or electron-donating effects. Meta directing groups generally withdraw electrons from the ring. Let's identify the meta directing groups among the given options:
−OCH3: This is an electron-donating group due to the strong +M (mesomeric) effect of the lone pair on oxygen, making it an ortho/para director.−NO2: This is a strongly electron-withdrawing group by resonance and inductive effects, making it a meta director.−CN: The cyano group is electron-withdrawing due to the presence of a highly electronegative nitrogen atom; hence, it is a meta director.−CH3: This is a weakly electron-donating group via hyperconjugation, making it an ortho/para director.−NHCOCH3: The amide linkage has resonance stabilization that donates electrons, making it an ortho/para director.−COR: This carbonyl functional group is electron-withdrawing due to the resonance and inductive effect, rendering it a meta director.−OH: Similar to methoxy, it is a strong electron-donating group due to the +M effect and thus an ortho/para director.−COOH: The carboxyl group is electron-withdrawing by resonance and induction, making it a meta director.−Cl: Though chlorine is electronegative, it is ortho/para directing because of its ability to donate electrons via resonance.Thus, the meta directing groups are: −NO2, −CN, −COR, and −COOH.
Counting these, there are 4 meta directing groups.
The meta-directing functional groups are those that are electron-withdrawing, which include:
\(-NO_2, -C \equiv N, -COR, -COOH\)


Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all \( x, y \in \mathbb{R}. \)
If \( f'(0) = \frac{1}{2} \), then the value of \( 24f''\left( \frac{5\pi}{3} \right) \) is: