Question

Suppose f is a function with the property that |f(x)| =< x^2

Show f(0)=f'(0)=0.

Answer 1

\[\left| f(x) \right|\le x^2\]\[-x^2\le f(x)\le x^2\]\[-2x \le f'(x) \le 2x\]\[0 \le f(0) \le 0\] and \[0 \le f'(0) \le 0\]
which means both f(0) and f'(0) are between 0 and 0, which can only happen when they EQUAL 0

Answer 2

Thank you!