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Question: If\(f(x) = {\cos ^2}x + {\cos ^2}2x + {\cos ^2}3x\), then number of values of \(x \in [0,2\pi ]\)for...

Iff(x)=cos⁡2x+cos⁡22x+cos⁡23xf(x) = {\cos ^2}x + {\cos ^2}2x + {\cos ^2}3x, then number of values of x∈[0,2π]x \in [0,2\pi ]for which f(x)=0f(x) = 0 are
A.4
B.6
C.8
D.0

Explanation

Solution

Hint: All the terms in the right hand side are squared. Think about what we can deduce from here.

Given, f(x)=cos⁡2x+cos⁡22x+cos⁡23xf(x) = {\cos ^2}x + {\cos ^2}2x + {\cos ^2}3x.
Since, every term in the right-hand side of the equation is squared, the value of each individual term could be either zero or greater than zero. We are interested in xx where f(x)=0f(x) = 0. Forf(x)f(x)to be zero, each individual on the right-hand side has to be zero. That iscos⁡2x=0,cos⁡22x=0 and cos⁡23x=0{\cos ^2}x = 0,{\cos ^2}2x = 0{\text{ and }}{\cos ^2}3x = 0.Now let’s solve them one by one.
cos⁡2x=0⇔cos⁡x=0⇔x=π2,3π2{\cos ^2}x = 0 \Leftrightarrow \cos x = 0 \Leftrightarrow x = \frac{\pi }{2},\frac{{3\pi }}{2}
cos⁡22x=0⇔cos⁡2x=0⇔2x=π2,3π2⇔x=π4,3π4{\cos ^2}2x = 0 \Leftrightarrow \cos 2x = 0 \Leftrightarrow 2x = \frac{\pi }{2},\frac{{3\pi }}{2} \Leftrightarrow x = \frac{\pi }{4},\frac{{3\pi }}{4}
cos⁡23x=0⇔cos⁡3x=0⇔3x=π2,3π2⇔x=π6,3π6{\cos ^2}3x = 0 \Leftrightarrow \cos 3x = 0 \Leftrightarrow 3x = \frac{\pi }{2},\frac{{3\pi }}{2} \Leftrightarrow x = \frac{\pi }{6},\frac{{3\pi }}{6}
Observe that, there is no common value of xx in all the above terms. The question should come in our mind as to why we are finding the common values. It’s just because we want xx wheref(x)=0f(x) = 0 and f(x)=0f(x) = 0 when all the individual terms on the right hand side will be zero. It means for a single value of xx, all the terms on the right-hand side has to vanish simultaneously. That’s why we are looking at the common value of xx where cos⁡2x=0,cos⁡22x=0 and cos⁡23x=0{\cos ^2}x = 0,{\cos ^2}2x = 0{\text{ and }}{\cos ^2}3x = 0.But, there is no such common value in the given domain. So, there is no xx for which f(x)=0f(x) = 0.
Hence the correct option is D

Note: When you are finding the roots of something, keep domain in your mind. Here we have given our domain as f(x)=0f(x) = 0. So, we only considered such x where f(x)=0f(x) = 0 in the given domain. One should not step out of the domain.