If `f(x)=|cos x|, then f' ((3pi)/(4))` is equal to

Options

A. `-1/sqrt2`
B. `1/sqrt2`
C. 1
D. None of these

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 \( \cos \left( \frac{3\pi}{4} \right) \)
First, we need to evaluate \( \cos \left( \frac{3\pi}{4} \right) \).
\[ \cos \left( \frac{3\pi}{4} \right) = -\frac{1}{\sqrt{2}} \]
Since \( \cos \left( \frac{3\pi}{4} \right) \) is negative, we can express \( f(x) \) in terms of \( -\cos x \).

### Step 2: Express \( f(x) \) for \( x = \frac{3\pi}{4} \)
Since \( \cos \left( \frac{3\pi}{4} \right) < 0 \), we have:
\[ f(x) = -\cos x \quad \text{for } x = \frac{3\pi}{4} \]

### Step 3: Differentiate \( f(x) \)
Now, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx} (-\cos x) = \sin x \]

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

### Step 5: Calculate \( \sin \left( \frac{3\pi}{4} \right) \)
We know that:
\[ \sin \left( \frac{3\pi}{4} \right) = \sin \left( \pi - \frac{\pi}{4} \right) = \sin \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, we have:
\[ f' \left( \frac{3\pi}{4} \right) = \frac{1}{\sqrt{2}} \]

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