If `y= 2 sin x+3 cosx`, then `y + (d^(2)y)/(dx^(2))` is

Options

A. 1
B. 0
C. 2
D. 3

Answer

To solve the problem, we need to find \( y + \frac{d^2y}{dx^2} \) given that \( y = 2 \sin x + 3 \cos x \).

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Given:
\[ y = 2 \sin x + 3 \cos x \]

Using the derivatives of sine and cosine:
\[ \frac{dy}{dx} = 2 \cos x - 3 \sin x \]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)

Now, we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2 \cos x - 3 \sin x) \]

Using the derivatives of cosine and sine:
\[ \frac{d^2y}{dx^2} = -2 \sin x - 3 \cos x \]

### Step 3: Substitute \( y \) and \( \frac{d^2y}{dx^2} \) into the expression \( y + \frac{d^2y}{dx^2} \)

Now we substitute the values of \( y \) and \( \frac{d^2y}{dx^2} \):
\[ y + \frac{d^2y}{dx^2} = (2 \sin x + 3 \cos x) + (-2 \sin x - 3 \cos x) \]

### Step 4: Simplify the expression

Combining the terms:
\[ y + \frac{d^2y}{dx^2} = 2 \sin x + 3 \cos x - 2 \sin x - 3 \cos x = 0 \]

### Final Answer

Thus, we find that:
\[ y + \frac{d^2y}{dx^2} = 0 \]

### Conclusion

The answer is \( 0 \), which corresponds to option \( b \).

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