Which one of the following is correct in respect of the curve `4y-x^(2)-8=0` ?

Options

A. The curve is increasing in (-4,4)
B. The curve is increasing (-4,0)
C. The curve is increasing in (0,4)
D. The curve is decreasing in (-4,4)

Explanation

To determine the behavior of the curve defined by the equation \(4y - x^2 - 8 = 0\), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the given equation to express \(y\) in terms of \(x\).

\[ 4y = x^2 + 8 \]
\[ y = \frac{x^2 + 8}{4} \]

### Step 2: Differentiate \(y\) with respect to \(x\)
Next, we will find the derivative \(\frac{dy}{dx}\) to analyze the increasing or decreasing nature of the curve.

\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{x^2 + 8}{4}\right) \]

Using the power rule for differentiation:

\[ \frac{dy}{dx} = \frac{1}{4} \cdot \frac{d}{dx}(x^2 + 8) = \frac{1}{4} \cdot (2x + 0) = \frac{2x}{4} = \frac{x}{2} \]

### Step 3: Determine intervals of increase and decrease
To find where the curve is increasing or decreasing, we need to analyze the sign of \(\frac{dy}{dx}\).

\[ \frac{dy}{dx} > 0 \quad \text{(increasing)} \]
\[ \frac{x}{2} > 0 \implies x > 0 \]

### Step 4: Conclusion
The curve is increasing when \(x > 0\). Therefore, we can conclude that the curve is increasing in the interval \( (0, \infty) \).