Show that y=1/x is concave up for x>0 and concave down for x<0.
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y " = (1/x) " = (-1/x ²) ' = 2/x³
for x = -1 ==>y"(-1) = - 2 <0 ==> from -∞ to 0 concave down
x = 1====> y"(1) = 2 >0 ===> concave up
for x = -1 ==>y"(-1) = - 2 <0 ==> from -∞ to 0 concave down
x = 1====> y"(1) = 2 >0 ===> concave up
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y = x^-1
y' = dy/dx = -x^-2
y'' = dy'/dx = 2*x^-3 = 2/x^3
y''(x>0) > 0, therefore concave up
y''(x<0) < 0, therefore concave down
y' = dy/dx = -x^-2
y'' = dy'/dx = 2*x^-3 = 2/x^3
y''(x>0) > 0, therefore concave up
y''(x<0) < 0, therefore concave down