The maximum value of the function f(x) = a si | vedclass

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Question

(A) To find the maximum value of the function $f(x) = a \sin x + b \cos x$,we can rewrite it in the form $R \sin(x + \alpha)$.We multiply and divide b

Vedclass · Accepted Answer

$\sqrt{a^2+b^2}$

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(A) To find the maximum value of the function $f(x) = a \sin x + b \cos x$,we can rewrite it in the form $R \sin(x + \alpha)$.We multiply and divide by $\sqrt{a^2 + b^2}$:$f(x) = \sqrt{a^2 + b^2} \left( \frac{a}{\sqrt{a^2 + b^2}} \sin x + \frac{b}{\sqrt{a^2 + b^2}} \cos x \right)$.Let $\cos \alpha = \frac{a}{\sqrt{a^2 + b^2}}$ and $\sin \alpha = \frac{b}{\sqrt{a^2 + b^2}}$.Then $f(x) = \sqrt{a^2 + b^2} (\sin x \cos \alpha + \cos x \sin \alpha) = \sqrt{a^2 + b^2} \sin(x + \alpha)$.Since the maximum value of $\sin(x + \alpha)$ is $1$,the maximum value of $f(x)$ is $\sqrt{a^2 + b^2}$.

The maximum value of the function $f(x) = a \sin x + b \cos x$ is:

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