Let `f(x)=e^x sinx` , slope of the curve y=f(x) is maximum at x=a if 'a' equals

Options

A. 0
B. `pi//4`
C. `pi//2`
D. none of these

Answer

To solve the problem, we need to find the value of \( a \) where the slope of the curve \( y = f(x) = e^x \sin x \) is maximum. The slope of the curve is given by the first derivative \( f'(x) \).

### Step-by-Step Solution:

1. **Find the first derivative \( f'(x) \)**:
\[ f(x) = e^x \sin x \]
Using the product rule:
\[ f'(x) = \frac{d}{dx}(e^x) \cdot \sin x + e^x \cdot \frac{d}{dx}(\sin x) \]
\[ f'(x) = e^x \sin x + e^x \cos x = e^x (\sin x + \cos x) \]

2. **Set \( f'(x) \) as \( p(x) \)**:
\[ p(x) = e^x (\sin x + \cos x) \]

3. **Find the second derivative \( p'(x) \)** to locate the maximum:
\[ p'(x) = \frac{d}{dx}[e^x (\sin x + \cos x)] \]
Using the product rule again:
\[ p'(x) = e^x (\sin x + \cos x) + e^x (\cos x - \sin x) \]
\[ p'(x) = e^x [(\sin x + \cos x) + (\cos x - \sin x)] = e^x (2 \cos x) \]

4. **Set \( p'(x) = 0 \)** to find critical points:
\[ e^x (2 \cos x) = 0 \]
Since \( e^x \neq 0 \) for all \( x \), we have:
\[ 2 \cos x = 0 \implies \cos x = 0 \]
The solutions to \( \cos x = 0 \) are:
\[ x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z} \]

5. **Determine which critical points give a maximum**:
We will check \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

6. **Find the second derivative \( p''(x) \)**:
\[ p''(x) = \frac{d}{dx}[e^x (2 \cos x)] \]
\[ p''(x) = e^x (2 \cos x) + e^x (-2 \sin x) = 2 e^x (\cos x - \sin x) \]

7. **Evaluate \( p''(x) \) at \( x = \frac{\pi}{2} \)**:
\[ p''\left(\frac{\pi}{2}\right) = 2 e^{\frac{\pi}{2}} (\cos(\frac{\pi}{2}) - \sin(\frac{\pi}{2})) = 2 e^{\frac{\pi}{2}} (0 - 1) = -2 e^{\frac{\pi}{2}} \]
Since \( p''\left(\frac{\pi}{2}\right) < 0 \), this indicates a local maximum.

8. **Check \( x = \frac{3\pi}{2} \)**:
\[ p''\left(\frac{3\pi}{2}\right) = 2 e^{\frac{3\pi}{2}} (\cos(\frac{3\pi}{2}) - \sin(\frac{3\pi}{2})) = 2 e^{\frac{3\pi}{2}} (0 + 1) = 2 e^{\frac{3\pi}{2}} \]
Since \( p''\left(\frac{3\pi}{2}\right) > 0 \), this indicates a local minimum.

### Conclusion:
The slope of the curve \( y = f(x) = e^x \sin x \) is maximum at:
\[ \boxed{\frac{\pi}{2}} \]