Maximum And Minimum Values

Question

For all real $x$, the maximum possible value of the function $f(x) = 3\sin x + 4\cos x + 7$ is determined by amplitude reasoning to be which of the following numbers?

RankGuru · Accepted Answer

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 $R\sin(x + \phi)$ 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.

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Maximum And Minimum Values

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Solution

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If $cos θ = \frac{3}{5}$ with $θ$ in the first quadrant, then applying the half-angle iden...

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Statement based: assertion that $sin^{- 1} (sin \frac{2 π}{3}) = \frac{2 π}{3}$ and reason that arc...

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