Find Intervals of Increase and Decrease for y = (1/√2)x² - 5

To solve this problem, we follow a methodical approach:

• Step 1: Compute the derivative of the function.
• Step 2: Find the critical points where the derivative is zero.
• Step 3: Determine the sign of the derivative on intervals split by the critical points.
• Step 4: Identify intervals of increase and decrease based on the sign of the derivative.

Now, let's work through each step:

Step 1: Given the function $y = \frac{1}{\sqrt{2}}x^2 - 5$, find its derivative:

Calculating the derivative: $y' = \frac{d}{dx} \left( \frac{1}{\sqrt{2}}x^2 - 5 \right) = \frac{2}{\sqrt{2}}x = \sqrt{2}x$.

Step 2: Set the derivative to zero to find the critical points:

$\sqrt{2}x = 0$ implies $x = 0$.

Step 3: Determine the sign of the derivative on the intervals $x < 0$ and $x > 0$:

• For $x < 0$, $\sqrt{2}x < 0$, so $y' < 0$ (the function is decreasing).
• For $x > 0$, $\sqrt{2}x > 0$, so $y' > 0$ (the function is increasing).

Step 4: Thus, the function is decreasing on the interval $x < 0$ and increasing on the interval $x > 0$.

Therefore, the solution to the problem is $\searrow:x<0\\\nearrow:x>0$.