concave up and concave down

Question

determine the open intervals on which the graph of the function is concave upward or concave downward. enter your answer in interval notation

 

f(x)= x^2/ (x^2+49)

Answer 1

f(x)=x2/(x2+49)=(x2+49-49)/(x2+49)=1-49/(x2+49).

f'(x)=-49(2x)/(x2+49)2=-98x/(x2+49)2.

The only turning point is when this expression is zero, that is, at x=0.

f(-1)=f(1)=1/50. So on either side of zero the curve is moving away from the x-axis, making it concave upward, with minimum at (0,0), the origin. When x is large negative f(x)→1 as -49/(x2+49)→0. When x is large positive f(x)→1 (asymptote). So for (-∞,0] f(x) is decreasing towards zero, while for [0,∞) it's increasing from zero.