The period of the function `f(x)=k cos kx ` is `(pi)/(2)`. The amplitude of f is

Options

A. `1/4`
B. `1/2`
C. 1
D. 4

Explanation

To solve the problem, we need to find the amplitude of the function \( f(x) = k \cos(kx) \) given that its period is \( \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Identify the standard form of the cosine function**:
The standard form of a cosine function is given by:
\[ f(x) = a \cos(bx) \]
where \( a \) is the amplitude and \( b \) is related to the period.

2. **Relate the period to the function**:
The period \( T \) of the function is given by the formula:
\[ T = \frac{2\pi}{b} \]
In our case, since \( f(x) = k \cos(kx) \), we can see that \( b = k \).

3. **Set up the equation for the given period**:
We know from the problem that the period is \( \frac{\pi}{2} \). Therefore, we can set up the equation:
\[ \frac{2\pi}{k} = \frac{\pi}{2} \]

4. **Solve for \( k \)**:
To find \( k \), we can cross-multiply:
\[ 2\pi \cdot 2 = \pi \cdot k \]
Simplifying this gives:
\[ 4\pi = \pi k \]
Dividing both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ k = 4 \]

5. **Determine the amplitude**:
From the standard form, we know that the amplitude \( a \) is equal to \( k \). Since we found \( k = 4 \), we have:
\[ \text{Amplitude} = a = k = 4 \]

### Final Answer:
The amplitude of the function \( f(x) = k \cos(kx) \) is \( 4 \).

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