Analyze Domain Behavior: Finding Increasing and Decreasing Intervals of f(x)=-2x²+10

To solve this problem, we'll identify the intervals where the function $f(x) = -2x^2 + 10$ is increasing and decreasing. Here’s how we can tackle it:

• Step 1: Locate the Vertex
  Since the function is $f(x) = -2x^2 + 10$, it follows the standard form of a quadratic: $ax^2 + bx + c$ where $a = -2$, $b = 0$, and $c = 10$. The vertex of this parabola lies at $x = -\frac{b}{2a}$. Plugging the values in, we find: \begin{align*} x &= -\frac{0}{2(-2)} \\ x &= 0 \end{align*} So, the vertex is located at $x = 0$.
• Step 2: Determine Increasing and Decreasing Intervals
  Since $a = -2$, which is negative, the parabola opens downward. This means the function will increase as it approaches the vertex and decrease after passing it. Therefore: \begin{enumerate}
• The function is increasing when $x < 0$.
• The function is decreasing when $x > 0$.
• Verification with Choices
  Comparing with the given choices, the correct answer is that the function is decreasing when $x > 0$ and increasing when $x < 0$.

Therefore, the intervals are:

$0 < x$ decreasing

$x < 0$ increasing