Verify if the function f(x) = xarctanx - ln(1+x^2) is concave upwards .

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If a function is concave upwards then it's second derivative is positive.

We'll determine the 1st derivative:

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

f'(x) = arctanx  –  x/(1+x^2)

We'll determine the 2nd derivative

f"(x)= 1/(1+x^2) – (1+x^2-2x^2)/(1+x^2)^2

f"(x) = (1+x^2+x^2-1)/(1+x^2)^2

We'll combine like terms inside brackets:

f"(x)=2x^2/(1+x^2) >= 0

Since the second derivative is always positive , for any real value of x, then the function is concave upwards.

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