The interval in which the function f given by `f(x) = 7 - 4x + x^(2)` is strictly decreasing, is :

Options

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

Answer

To determine the interval in which the function \( f(x) = 7 - 4x + x^2 \) is strictly decreasing, we will follow these steps:

### Step 1: Differentiate the function
First, we need to find the derivative of the function \( f(x) \).

\[ f(x) = 7 - 4x + x^2 \]

Differentiating \( f(x) \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx}(7) - \frac{d}{dx}(4x) + \frac{d}{dx}(x^2) \]

Calculating the derivatives:

\[ f'(x) = 0 - 4 + 2x = 2x - 4 \]

### Step 2: Set the derivative less than zero
To find where the function is strictly decreasing, we need to set the derivative \( f'(x) \) less than zero:

\[ f'(x) < 0 \]

Substituting our expression for \( f'(x) \):

\[ 2x - 4 < 0 \]

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

\[ 2x < 4 \]

Dividing both sides by 2:

\[ x < 2 \]

### Step 4: Write the interval
The function \( f(x) \) is strictly decreasing for all \( x \) values less than 2. In interval notation, this can be expressed as:

\[ (-\infty, 2) \]

### Conclusion
Thus, the interval in which the function \( f(x) \) is strictly decreasing is:

\[ \boxed{(-\infty, 2)} \]

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