How do you use the closed-interval method to find the absolute maximum and minimum values of the function #f(x)=x-2sinx# on the interval #[-π/4, π/2]#?
2 Answers
The answer is:
First of all, let's see if there are some local maximum o minimum in that interval.
So the function grows in
and decreases in
graph{x-2sinx [-4.93, 4.94, -2.465, 2.466]}
So the point
Now we have to calculate the ordinate of both the extremes of the interval, because the absolute maximum and minimum could be in that points.
The points are:
So the absolute maximum is
First of all, let's recall how the method works: if you have a continuous function (which is
So, first of all, let's derive the function: since the derivative of a sum is the sum of the derivatives, you have that
Now, let's recall that we can factor out constants:
These are both elementary derivatives, and we have
This function has zeroes if and only if
The only thing left is thus to compare the values of the function in
Some easy computations show that:
And so the minimum value of