Which one of the following statements 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 in (-4, 0)
C. The curve is increasing in (0,4)
D. The curve is decreasing in (-4, 4)

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the equation to express \(y\) in terms of \(x\):
\[ 4y - x^2 - 8 = 0 \implies 4y = x^2 + 8 \implies y = \frac{x^2}{4} + 2 \]

### Step 2: Finding the Derivative
Next, we differentiate \(y\) with respect to \(x\) to find \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{x^2}{4} + 2\right) = \frac{2x}{4} = \frac{x}{2} \]

### Step 3: Analyzing the Derivative
Now, we analyze the sign of the derivative \(\frac{dy}{dx}\):
- \(\frac{dy}{dx} = \frac{x}{2}\)

This derivative will be:
- Positive when \(x > 0\)
- Zero when \(x = 0\)
- Negative when \(x < 0\)

### Step 4: Determining Intervals of Increase and Decrease
From the analysis of the derivative:
- The function is **decreasing** on the interval \((- \infty, 0)\) because \(\frac{dy}{dx} < 0\).
- The function is **increasing** on the interval \((0, \infty)\) because \(\frac{dy}{dx} > 0\).

### Step 5: Conclusion
Based on the intervals we found:
- The curve is **decreasing** in the interval \((- \infty, 0)\) and **increasing** in the interval \((0, \infty)\).

### Final Answer
Thus, the correct statement regarding the curve \(4y - x^2 - 8 = 0\) is that it is decreasing in \((- \infty, 0)\) and increasing in \((0, \infty)\).

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