Consider the trigonometric function on the interval (0,2pi). Find the open intervals on which the function is increasing or decreasing.
My first questions is
1.f(x)=sin^2(x)+sin(x)
2 sin(x) cos(x)+cos(x)
cos(x) 2sin(x)+1=0 (factor)
sin(x)=-1/2
x=7pi/6,11pi/6 Did I do this correctly
My second question is
f(x)=(x-1)^(2/3)
(2/3)(x-1)^(-1/3)(1)=0
How would I solve this one I seem to be getting an incorrect answer
critical values of a function occur where the function's derivative equals zero or is undefined at x-values in the function's domain.
the whole point here is that if a function has extrema on an open interval, then they occur at critical values. note that this is not a two-way street ... a function that has critical values may not have extrema located there.
the first derivative test procedure is to check the sign of the derivative on either side of the critical value. if the sign of f' changes from positive to negative, then the original function has a maximum there ... if f' changes from negative to positive, then the original function has a minimum there. sometimes the sign will not change at a critical value ... no extrema at that value.
your second function is an example of an extremum located where f'(x) is undefined ...
note ... the function is defined.
also note that for any value of x , but is undefined at x = 1.
for x < 1 , f'(x) < 0 ... f(x) is decreasing
for x > 1 , f'(x) > 0 ... f(x) is increasing
f(x) has a minimum at x = 1 ... see the graph for confirmation.