Please check my Caclulus

1. Find all intervals on which the graph of y=(x^2+1)/x^2 is concave upward.
A. (negative infinity, infinity)
B. (negative infinity, -1) U (1, infinity)
C. (negative infinity, 0) U (0, infinity)
D. (1, infinity)
E. none of these
I got C. I found the second derivative and used the interval test.

2. If f(3)=0, f’(3)=6, g(3)=1, g’(3)=1/3, find h’(3) if h(x)=[f(x)] / [g(x)]
A. 18
B. 6
C. -6
D. -2
E. none of these
I got B. I figured out h'(x) in terms of f(x), g(x), f'(x), and g'(x) and plugged in the given numbers.

3. Find all open intervals on which f(x)=x/(x^2+x-2) is decreasing.
A. (negative infinity, infinity)
B. (negative infinity, 0)
C. (negative infinity, -2) U (1, infinity)
D. (negative infinity, -2) U (-2,1) U (1, infinity)
E. none of these
I got C. I found the first derivative and critical numbers. Then I used the interval test.

Thank you for checking my answers.

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  1. #1 ok

    #2
    h' = (f'g-fg')/g^2 = (6*1-0*1/3)/1 = 6
    So, B is correct

    #3
    f' = -(x^2+2)/(x^2+x-2)^2
    f' < 0 for all x
    So, (D) is correct

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