The function f(x) =`x^(2)-2x` increase forall

Options

A. `xgt-1` only
B. `x lt =1` only
C. `x gt 1` only
D. `x lt 1` only

Explanation

To determine the intervals where the function \( f(x) = x^2 - 2x \) is increasing, we will follow these steps:

### Step 1: Find the derivative of the function
The first step is to find the derivative of the function \( f(x) \).

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

Using the power rule, we differentiate:

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

### Step 2: Set the derivative greater than zero
For the function to be increasing, the derivative must be greater than zero:

\[ f'(x) > 0 \]

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

\[ 2x - 2 > 0 \]

### Step 3: Solve the inequality
Now, we will solve the inequality \( 2x - 2 > 0 \):

\[ 2x > 2 \]

Dividing both sides by 2:

\[ x > 1 \]

### Step 4: Conclusion
The function \( f(x) = x^2 - 2x \) is increasing for \( x > 1 \).

### Final Answer
The function is increasing for \( x > 1 \).

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