Range of `f(x)=log_(sqrt10) (sqrt5 (2 sin x+cos x)+5)` is :

Options

A. [0,1]
B. [0,3]
C. `(-oo,1/3]`
D. None of these

Answer

To find the range of the function \( f(x) = \log_{\sqrt{10}} \left( \sqrt{5} (2 \sin x + \cos x) + 5 \right) \), we will follow these steps:

### Step 1: Determine the range of \( 2 \sin x + \cos x \)

The expression \( 2 \sin x + \cos x \) can be analyzed using the formula for the range of a linear combination of sine and cosine functions. The range of \( a \sin x + b \cos x \) is given by \( [-\sqrt{a^2 + b^2}, \sqrt{a^2 + b^2}] \).

Here, \( a = 2 \) and \( b = 1 \).

\[ \text{Range of } 2 \sin x + \cos x = \left[-\sqrt{2^2 + 1^2}, \sqrt{2^2 + 1^2}\right] = \left[-\sqrt{5}, \sqrt{5}\right] \]

### Step 2: Transform the range to \( \sqrt{5} (2 \sin x + \cos x) + 5 \)

Next, we will transform the range we found in step 1 by multiplying by \( \sqrt{5} \) and then adding \( 5 \).

1. Multiply the range \( [-\sqrt{5}, \sqrt{5}] \) by \( \sqrt{5} \):

\[ \text{New range} = \left[-\sqrt{5} \cdot \sqrt{5}, \sqrt{5} \cdot \sqrt{5}\right] = [-5, 5] \]

2. Now add \( 5 \) to the entire range:

\[ \text{Final range} = [-5 + 5, 5 + 5] = [0, 10] \]

### Step 3: Apply the logarithm function

Now we need to consider the function \( f(x) = \log_{\sqrt{10}}(y) \) where \( y \) is in the range \( [0, 10] \).

- The logarithm function \( \log_{\sqrt{10}}(y) \) is defined for \( y > 0 \).
- The minimum value of \( y \) is \( 0 \) (but not included, since log is undefined at 0).
- The maximum value of \( y \) is \( 10 \).

### Step 4: Evaluate the logarithm at the endpoints

1. As \( y \) approaches \( 0 \), \( \log_{\sqrt{10}}(y) \) approaches \( -\infty \).
2. Evaluate \( \log_{\sqrt{10}}(10) \):

\[ \log_{\sqrt{10}}(10) = \frac{\log_{10}(10)}{\log_{10}(\sqrt{10})} = \frac{1}{\frac{1}{2}} = 2 \]

### Conclusion: Determine the range of \( f(x) \)

Thus, the range of \( f(x) \) is:

\[ (-\infty, 2) \]

### Final Answer

The range of \( f(x) = \log_{\sqrt{10}} \left( \sqrt{5} (2 \sin x + \cos x) + 5 \right) \) is \( (-\infty, 2) \).

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