Hey all, could someone please check my answers? thanks
1.) Consider the function k(x) = (x^2+1)arctan(x)
k'(x) = 1+2xarctan(x)
k''(x) = (2x/x^2+1)+ 2arctan(x)
a) Find the domain of k
my solution: dom(k) = R
b) Find x and y intercepts
my solution x int at (0,0) and y int at (0,0)
c) Find the interval on which k is increasing:
my solution: since k'(x) > 0 on (0,inf), k increases from (0,inf)
d) Find the interval on which k is decreasing
my solution: since k'(x) < 0 on (-inf,0), k decreases from (-inf, 0)
e) Does k have any stationary points?
my solution: k'(x) = 0
1+2xarctan(x) = 0
arctan(x) = -(1/2x)
since the two graphs never intersect, there are no stationary points
f) show that k has point of inflection at x = 0
my solution: k''(0) = 0, therefore point of inflection.
g) Is the point of inflection at x = 0 stationary or non stationary?
my solution: k'(0) = 1, therefore non stationary point of inflection at x = 0
h) Does k have any other point of inflections?
my solution: k''(x) = 0
(2x/1+x^2)+2 arctan(x) = 0
I got stuck at this point, any guidance would be much appreciated.