Minimum value of `3 sin theta+4 cos theta` in the interval `[0, (pi)/(2)]` is :

Options

A. -5
B. 3
C. 4
D. `(7)/(sqrt(2))`

Answer

To find the minimum value of the expression \(3 \sin \theta + 4 \cos \theta\) in the interval \([0, \frac{\pi}{2}]\), we can follow these steps:

### Step 1: Understand the function
The expression we are dealing with is \(f(\theta) = 3 \sin \theta + 4 \cos \theta\). We need to find its minimum value in the given interval.

### Step 2: Evaluate the endpoints
We will first evaluate the function at the endpoints of the interval, which are \(\theta = 0\) and \(\theta = \frac{\pi}{2}\).

- For \(\theta = 0\):
\[ f(0) = 3 \sin(0) + 4 \cos(0) = 3 \cdot 0 + 4 \cdot 1 = 4 \]

- For \(\theta = \frac{\pi}{2}\):
\[ f\left(\frac{\pi}{2}\right) = 3 \sin\left(\frac{\pi}{2}\right) + 4 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 1 + 4 \cdot 0 = 3 \]

### Step 3: Compare the values
Now we compare the values obtained:
- \(f(0) = 4\)
- \(f\left(\frac{\pi}{2}\right) = 3\)

### Step 4: Determine the minimum value
From our evaluations, we see that the minimum value occurs at \(\theta = \frac{\pi}{2}\):
\[ \text{Minimum value} = 3 \]

### Conclusion
Thus, the minimum value of \(3 \sin \theta + 4 \cos \theta\) in the interval \([0, \frac{\pi}{2}]\) is \(\boxed{3}\).

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