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AP EAMCET · Maths · Functions

The maximum and minimum values of the function \(f:[R \rightarrow[R\) defined by \(f(x)=5 \cos x+3 \cos \left(x+\frac{\pi}{3}\right)+8\) for all \(x \in[R\), are respectively.

  1. A 15, 1
  2. B 8, - 8
  3. C -7, - 15
  4. D 1, - 15
Verified Solution

Answer & Solution

Correct Answer

(A) 15, 1

Step-by-step Solution

Detailed explanation

\begin{aligned} & f(x)=5 \cos x+3 \cos \left(x+\frac{\pi}{3}\right)+8 \\ & =5 \cos x+3\left[\cos x \cdot \cos \frac{\pi}{3}-\sin x \sin \frac{\pi}{3}\right]+8 \\ & =5 \cos x+3\left[\frac{1}{2} \cos x-\frac{\sqrt{3}}{2} \sin x\right]+8 \\ & =\frac{13}{2} \cos x-\frac{3…