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Answer 1 · 2018-01-31T20:58:52

See explanation.

To check if the function is increasing or decreasing we have to calculate its first derivative.

#f(x)=2/x+x/2#

#f'(x)=(2/x)'+1/2#

#f'(x)=(2x^-1)'+1/2#

#f'(x)=(2x^-2)/(-2)+1/2#

#f'(x)=-1/x^2+1/2#

graph{-1/x^2+1/2 [-10, 10, -5, 5]}

From the graph of the derivative we see that:

#f'(x)>0 iff x in (-oo;-sqrt(2))uu(sqrt(2);+oo)#

#f'(x)<0 iff x in (-sqrt(2);0)uu(0;sqrt(2))#

So we can say that:

#f(x)# is increasing if #x in (-oo;-sqrt(2))uu(sqrt(2);+oo)#

#f(x)# is decreasing if #x in (-sqrt(2);0)uu(0;sqrt(2))#