Answer:
The absolute maximum is about -5.84 at <em>x</em> = -2.
And the absolute minimum is about -11.28 at <em>x</em> = -π/6.
Step-by-step explanation:
We want to find the absolute maximum and minimum values of the function:
First, we should evaluate the endpoints of the interval:
And:
Recall that extrema of a function occurs at its critical points. The critical points of a function are whenever its derivative is zero or undefined.
So, find the derivative of the function:
Differentiate:
Set the function equal to zero:
And solve for <em>x: </em>
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Using the unit circle, our solutions are:
There is only one solution in the interval [-2, 0]:
Thus, we only have one critical point on the interval.
Substituting this back into the function yields:
In conclusion, the absolute maximum value of <em>f</em> on the interval [-2, 0] is about -5.8385 at <em>x</em> = -2 and the absolute minimum value of <em>f</em> is about -11.2782 at <em>x</em> = -π/6.
We can see this from the graph below as well.