The least value of `5^(sinx-1)+5^(-sinx-1)`, is

Options

A. 10
B. `(5)/(2)`
C. `(2)/(5)`
D. `(1)/(5)`

Answer

To find the least value of the expression \(5^{(\sin x - 1)} + 5^{(-\sin x - 1)}\), we can follow these steps:

### Step 1: Rewrite the Expression
We start by rewriting the expression:
\[ y = 5^{(\sin x - 1)} + 5^{(-\sin x - 1)} \]
This can be simplified to:
\[ y = \frac{5^{\sin x}}{5} + \frac{1}{5^{\sin x} \cdot 5} \]
Thus,
\[ y = \frac{1}{5} \left( 5^{\sin x} + \frac{1}{5^{\sin x}} \right) \]

### Step 2: Set a New Variable
Let \(t = 5^{\sin x}\). Then the expression becomes:
\[ y = \frac{1}{5} \left( t + \frac{1}{t} \right) \]

### Step 3: Apply the AM-GM Inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we know that:
\[ \frac{t + \frac{1}{t}}{2} \geq 1 \]
This implies:
\[ t + \frac{1}{t} \geq 2 \]

### Step 4: Substitute Back into the Expression
Substituting back into our expression for \(y\):
\[ y \geq \frac{1}{5} \cdot 2 = \frac{2}{5} \]

### Step 5: Conclusion
Thus, the least value of \(y\) is:
\[ \frac{2}{5} \]

### Final Answer
The least value of \(5^{(\sin x - 1)} + 5^{(-\sin x - 1)}\) is \(\frac{2}{5}\).

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