The function `f(x) = x^(2) - 2 x` is strictly decreasing in the interval

Options

A. `(- infty, 1]`
B. `[1, infty)`
C. `[ - 1, infty)`
D. none of these

Answer

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

### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first need to find its derivative. The derivative \( f'(x) \) gives us the rate of change of the function.

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

Using the power rule, we differentiate:

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

### Step 2: Set the derivative less than zero
For the function to be strictly decreasing, we need to find where the derivative is less than zero:

\[ f'(x) < 0 \]

Substituting the expression we found:

\[ 2x - 2 < 0 \]

### Step 3: Solve the inequality
Now, we will solve the inequality:

\[ 2x < 2 \]

Dividing both sides by 2:

\[ x < 1 \]

### Step 4: Determine the interval
The inequality \( x < 1 \) indicates that the function is strictly decreasing for all values of \( x \) that are less than 1. Therefore, the interval where the function is strictly decreasing is:

\[ (-\infty, 1) \]

### Conclusion
Thus, the function \( f(x) = x^2 - 2x \) is strictly decreasing in the interval \( (-\infty, 1) \).

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