(a) Find the intervals in R for which the function f(x)=x4−2x2 is increasing or decreasing. OR (b) Find : ∫18−4x−x22x+1dx.
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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.