Maximum And Minimum Values

The amplitude form of a linear combination $a\sin x+b\cos x$ governs this problem, because any such combination can be rewritten as a single sinusoid $R\sin (x+\varphi )$ whose amplitude is $R=\sqrt{a^2+b^2}$, giving the range $[-R,R]$. The constant added on top simply shifts this range vertically without changing its width. Here the coefficients are $a=3$ and $b=4$, so the amplitude is $\sqrt{3^2+4^2}=\sqrt{25}=5$. Consequently the variable part $3\sin x+4\cos x$ oscillates between $-5$ and $5$, and adding the constant $7$ shifts the whole range to $[2,12]$. The maximum value of the function is therefore $7+5=12$. Option 14 wrongly uses the naive sum $a+b=7$ in place of the amplitude. Option 10 adds only part of the amplitude to the constant. Option 11 misadds the shift. This is the standard $a\sin x+b\cos x$ extremum pattern in JEE Advanced. As a final consistency check, the maximum occurs when $\sin x=\frac{3}{5}$ and $\cos x=\frac{4}{5}$, which satisfy ${\sin }^{2}x+{\cos }^{2}x=1$, confirming the amplitude of $5$ is attainable.