In which one of the following intervals is the function `f(x)=x^(2)-5x+6` decreasing?

Options

A. `(-oo,2]`
B. `[3,oo]`
C. `(-oo,oo)`
D. (2,3)

Explanation

To determine the interval in which the function \( f(x) = x^2 - 5x + 6 \) is decreasing, we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(x^2 - 5x + 6) \]

Using the power rule, we differentiate each term:

\[ f'(x) = 2x - 5 \]

### Step 2: Set the derivative less than zero
To find where the function is decreasing, we need to set the derivative less than zero:

\[ f'(x) < 0 \]

Substituting the expression we found for \( f'(x) \):

\[ 2x - 5 < 0 \]

### Step 3: Solve the inequality
Now, we solve the inequality for \( x \):

\[ 2x < 5 \]
\[ x < \frac{5}{2} \]

### Step 4: Determine the interval
The inequality \( x < \frac{5}{2} \) tells us that the function \( f(x) \) is decreasing for all values of \( x \) that are less than \( \frac{5}{2} \). Therefore, the interval in which the function is decreasing is:

\[ (-\infty, \frac{5}{2}) \]

### Step 5: Check the options
If we have options provided, we can check which one corresponds to the interval \( (-\infty, \frac{5}{2}) \).

### Conclusion
The function \( f(x) = x^2 - 5x + 6 \) is decreasing in the interval \( (-\infty, \frac{5}{2}) \).

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