(a) Find the intervals in $\mathbb{R}$ for which the function $f(x) = x^4 - 2x^2$ is increasing or decreasing. OR (b) Find : $\int \frac{2x + 1}{\sqrt{18 - 4x - x^2}} dx$ .

Answer: The function $f(x) = x^4 - 2x^2$ is increasing on the intervals $(-1, 0) \cup (1, \infty)$ and decreasing on the intervals $(-\infty, -1) \cup (0, 1)$ .

Step-by-step solution

Step 1: Differentiate the function

To determine where the function $f(x)$ is increasing or decreasing, we first need to find its derivative, $f'(x)$ . The derivative tells us the slope of the tangent line to the function at any point. If $f'(x) > 0$ , the function is increasing; if $f'(x) < 0$ , the function is decreasing.

Step 2: Find critical points

Next, we find the critical points by setting the derivative equal to zero and solving for $x$ . These points are where the function's slope is zero, indicating potential changes from increasing to decreasing or vice versa. The critical points divide the number line into intervals.

Step 3: Test intervals for increasing/decreasing behavior

We use the critical points to define intervals on the number line. Then, we pick a test point within each interval and substitute it into the derivative $f'(x)$ . The sign of $f'(x)$ in each interval tells us whether the function is increasing (positive) or decreasing (negative) in that interval.

Step 4: State the intervals

Based on the signs of the derivative in each interval, we can now state the intervals where the function is increasing and decreasing.

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(a) Find the intervals in R\mathbb{R}R for which the function f(x)=x4−2x2f(x) = x^4 - 2x^2f(x)=x4−2x2 is increasing or decreasing. OR (b) Find : ∫2x+118−4x−x2dx\int \frac{2x + 1}{\sqrt{18 - 4x - x^2}} dx∫18−4x−x2​2x+1​dx.