Find the greatest value of the function `f(x)=(2)/sqrt(2x^(2)-4x+3)`

Answer

To find the greatest value of the function \( f(x) = \frac{2}{\sqrt{2x^2 - 4x + 3}} \), we can follow these steps:

### Step 1: Analyze the function
The function \( f(x) \) is defined as \( f(x) = \frac{2}{\sqrt{g(x)}} \), where \( g(x) = 2x^2 - 4x + 3 \). To maximize \( f(x) \), we need to minimize \( g(x) \) since \( f(x) \) is inversely related to \( g(x) \).

### Step 2: Find the minimum of \( g(x) \)
To find the minimum of \( g(x) = 2x^2 - 4x + 3 \), we can complete the square or use calculus. Here, we will complete the square:

\[ g(x) = 2(x^2 - 2x) + 3 \]
Now, complete the square for \( x^2 - 2x \):

\[ x^2 - 2x = (x - 1)^2 - 1 \]
Substituting this back, we have:

\[ g(x) = 2((x - 1)^2 - 1) + 3 = 2(x - 1)^2 - 2 + 3 = 2(x - 1)^2 + 1 \]

### Step 3: Determine the minimum value of \( g(x) \)
The expression \( 2(x - 1)^2 + 1 \) reaches its minimum when \( (x - 1)^2 = 0 \), which occurs at \( x = 1 \). Thus, the minimum value of \( g(x) \) is:

\[ g(1) = 2(0) + 1 = 1 \]

### Step 4: Substitute the minimum value back into \( f(x) \)
Now that we have the minimum value of \( g(x) \), we can find the maximum value of \( f(x) \):

\[ f(x) = \frac{2}{\sqrt{g(x)}} \implies f(1) = \frac{2}{\sqrt{1}} = 2 \]

### Conclusion
The greatest value of the function \( f(x) \) is \( \boxed{2} \).

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