Identify the critical points and find the extreme values on the interval [-1,-5) for f(x)=cosx+xsinx+3
I've taken the derivative which gives me f'(x)=xcos(x). I know I have to solve to get the critical points and then plug in the critical points that are on the interval and the interval points to get the extreme values. I'm just not sure how to do those parts with this particular function. Please help
The critical points are where f' is zero or undefined. It is never undefined. So, remembering your Algebra I,
f' = 0 when x=0
or cosx=0: x = odd multiples of ?/2
x=0 is not in the domain, so the only critical points are at x = -?/2 and -3?/2.
Naturally, the extreme values also occur there.
http://www.wolframalpha.com/input/?i=cosx%2Bxsinx%2B3+for+-5+%3C+x+%3C%3D+-1
To find the critical points of a function, you'll need to solve for the values of x where the derivative is equal to zero or undefined. In this case, the derivative of f(x) is f'(x) = xcos(x).
Setting f'(x) equal to zero, we have:
xcos(x) = 0
To find the critical points, we need to analyze the values of x for which cos(x) = 0. The values of x that satisfy this equation are x = π/2 + nπ where n is an integer.
For x = π/2 + nπ, the values of f(x) on the interval [-1, -5) will only include f(-3π/2), f(-5π/2), and so on.
Now that we have found the critical points on the interval, we need to check the function values at these points and the endpoints of the interval to determine if they correspond to relative extrema.
To find extreme values, we evaluate the function at the critical points and at the endpoints of the interval:
f(-1) = cos(-1) + (-1)sin(-1) + 3,
f(-3π/2) = cos(-3π/2) + (-3π/2)sin(-3π/2) + 3,
f(-4) = cos(-4) + (-4)sin(-4) + 3,
and so on.
Once you have the function values for each point, compare them to determine if they correspond to maximum or minimum values on the interval [-1, -5).
The largest value among these function values will be the maximum on the given interval, and the smallest value will be the minimum.