Problem 10 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of values of \(x\), where the function \(f(x)=\) \(\cos x+\cos (\sqrt{2} x)\) attains its maximum, is (A) 0 (B) 1 (C) 2 (D) infinite

Step-by-Step Solution

Answer

The number of values is 0, option (A).

Step 1: Understand the Function

We need to determine the number of values of \(x\) for which the function \(f(x) = \cos x + \cos(\sqrt{2} x)\) attains its maximum value. The maximum value of a cosine function is 1.

Step 2: Determine Maximum Value

To find the maximum value of \(f(x)\), we should derivatively see if both cosine terms can reach 1 simultaneously. The expression hits a theoretical maximum when both \(\cos x = 1\) and \(\cos(\sqrt{2} x) = 1\).

Step 3: Conditions for Maximum

We set the conditions where \(\cos x = 1\) and \(\cos(\sqrt{2} x) = 1\). This requires \(x = 2k\pi\) and \(\sqrt{2}x = 2m\pi\), where \(k\) and \(m\) are integers.

Step 4: Solve Simultaneously for x

Using the previous conditions, we have \(x = 2k\pi\) and \(x = \frac{2m\pi}{\sqrt{2}} = \sqrt{2}m\pi\). For these to occur at the same time, \(\sqrt{2}m = 2k\) must hold.

Step 5: Check Integer Solution Feasibility

For \(\sqrt{2}m = 2k\), integer values of \(m\) and \(k\) do not yield a matching \(x\), due to the irrational nature of \(\sqrt{2}\). Hence, no integer solution exists for both conditions simultaneously.

Key Concepts

Cosine Function

The Cosine Function is a fundamental component in trigonometry. It is often represented by \(\cos(x)\) and measures the horizontal component of an angle in a unit circle. This trigonometric function oscillates between -1 and 1, creating a wave-like pattern. Its graph displays periodic behavior, repeating every \(2\pi\) radians.

• At \(x = 0, \pi, 2\pi,...\), the cosine takes a value of 1, which is the maximum.
• At \(x = \pi/2, 3\pi/2,...\), the cosine equals 0.
• At \(x = \-\pi/2, \pi, 3\pi/2,...\), the cosine takes on a value of -1.

Understanding these values helps in predicting the behavior of more complex functions that include cosine, such as our problem here, \(f(x) = \cos x + \cos(\sqrt{2} x)\). The function \(\cos(\sqrt{2} x)\) involves stretching the regular cosine function by a factor of \sqrt{2}\, which results in an altered frequency.

Maximum Value

The maximum value of any cosine function is 1. When combining functions like \(\cos x + \cos(\sqrt{2} x)\), to find the overall maximum, both components must independently reach their peaks.To achieve this, each component of the expression must equal 1 simultaneously.

• This happens when \(\cos x = 1\).
• Similarly, \(\cos(\sqrt{2} x) = 1\).

These situations occur when their respective angles are integer multiples of \(2\pi\). This involves setting equations where the angles in both functions must align to multiples of \(2\pi\), signifying their peaks, which bring us to specific conditions, \(x = 2k\pi\) and \(\sqrt{2} x = 2m\pi\).
This synchronization of maxima mathematically would require solving \(\sqrt{2}m = 2k\), thereby examining if overlapping peaks align adequately.

Irrational Numbers

Irrational numbers, like \(\sqrt{2}\), do not settle into neat ratios of integers. They are non-terminating and non-repeating when expressed in decimal form. While many numbers in trigonometry are rational, such as multiples of \(\pi\), irrational numbers provide infinite precision and beauty to mathematical expressions. The complexity of including an irrational component like \(\sqrt{2} x\) in \(\cos(\sqrt{2} x)\) is noteworthy. The \(\sqrt{2}\) stretches the cosine wave's frequency beyond intuitive cycles of \(2\pi\).

• When we attempt to align \(x = 2k\pi\) and \(\sqrt{2} x = 2m\pi\), the irrationality disrupts harmony.
• This lack of integer solutions for \(\sqrt{2}m = 2k\) means there's no shared solution for maximum values occurring simultaneously with integer precision.

Thus, directly finding the simultaneous maximum for our original \(f(x)\) is impractical, giving insight into why no solution meets the overlapping condition perfectly.