Find the relative extrema of on the interval
To calculate: The relative extrema of the trigonometric function over the interval .
The provided trigonometric function is over the interval .
Sine derivative Rule:
Cosine derivative Rule:
General power derivative rule:
Double angle formula for trigonometric identity.
Condition for critical points:
If function is increasing, then and if function is decreasing, then .
From first derivative Test, if changes positive to negative, then function has relative maxima and, if changes negative to positive then function has relative minima. In order words, if function changes from increasing function to decreasing at any point critical, then function will have relative maxima at that point, and if function changes from decreasing function to increasing at any critical point, then function will have relative minima at that point.
Consider the provided trigonometric function:
Differentiate the above function with respect to x using Sine, Cosine and Power derivative rules.
Apply in order to find critical points.
Since, Sine is positive in first and second quadrants, while, sine is negative in third and fourth quadrants. Thus, three are three solutions for over the interval , which can be mathematically calculated as:
Thus, critical points are , and
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