if `f(x) = | cos x| ` then ` f' ((3pi )/(4))` is

Options

A. `-(1)/( sqrt(2))`
B. `(1)/( sqrt(2))`
C. 1
D. None

Answer

To find \( f' \left( \frac{3\pi}{4} \right) \) for the function \( f(x) = |\cos x| \), we will follow these steps:

### Step 1: Determine the value of \( \cos \left( \frac{3\pi}{4} \right) \)
The angle \( \frac{3\pi}{4} \) is in the second quadrant. In the second quadrant, the cosine function is negative.

\[ \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} \]

### Step 2: Rewrite the function \( f(x) \)
Since \( \cos \left( \frac{3\pi}{4} \right) \) is negative, we can express \( f(x) \) as follows:

\[ f(x) = |\cos x| = -\cos x \quad \text{for } x = \frac{3\pi}{4} \]

### Step 3: Differentiate \( f(x) \)
Now we differentiate \( f(x) \). Since \( f(x) = -\cos x \), we can find the derivative:

\[ f'(x) = -\frac{d}{dx}(\cos x) = \sin x \]

### Step 4: Evaluate \( f' \left( \frac{3\pi}{4} \right) \)
Now we substitute \( x = \frac{3\pi}{4} \) into the derivative:

\[ f' \left( \frac{3\pi}{4} \right) = \sin \left( \frac{3\pi}{4} \right) \]

The angle \( \frac{3\pi}{4} \) corresponds to \( 135^\circ \), and in the second quadrant, the sine function is positive:

\[ \sin \left( \frac{3\pi}{4} \right) = \sin \left( 180^\circ - 45^\circ \right) = \sin 45^\circ = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the value of \( f' \left( \frac{3\pi}{4} \right) \) is:

\[ f' \left( \frac{3\pi}{4} \right) = \frac{1}{\sqrt{2}} \]

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