The interval in which the function f (x) = 7 `-4x - x ^(2)` is strictly increasing is :

Options

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

Answer

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

### Step 1: Find the derivative of the function
To find where the function is increasing, we first need to calculate the derivative \( f'(x) \).

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

Differentiating with respect to \( x \):

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

Since the derivative of a constant is 0, the derivative of \( 4x \) is 4, and the derivative of \( x^2 \) is \( 2x \):

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

### Step 2: Set the derivative greater than zero
For the function to be strictly increasing, we need \( f'(x) > 0 \):

\[ -4 - 2x > 0 \]

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

\[ -2x > 4 \]

Dividing both sides by -2 (remember to flip the inequality sign):

\[ x < -2 \]

### Step 4: Determine the interval
The function \( f(x) \) is strictly increasing for \( x < -2 \). Therefore, the interval in which the function is strictly increasing is:

\[ (-\infty, -2) \]

### Final Answer
The interval in which the function \( f(x) = 7 - 4x - x^2 \) is strictly increasing is \( (-\infty, -2) \).

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