Mathematics

Problem

Let

S

be a differentiable function with derivative

S'(x)=\frac{x^{2}}{1+x^{4}}

for all real

x

. Suppose

S(0)=0

. (a) Find the exact value of

S(2)

. (b) Evaluate

\displaystyle \lim_{x\to\infty} S(x)

, showing that the limit exists and stating its exact value. (c) Based on your work in parts (a) and (b) and the properties of

S'(x)

, determine whether

S

is bounded above, bounded below, both, or neither on the interval

[0,\infty)

. Provide a complete justification that references relevant limits and monotonicity considerations.