In fact, the process of this problem is

(1) derivative

(2) judging whether the derivative function f'(x) is positive or negative, so as to determine the increase or decrease of the original function f(x)

This is also the general process of judging the increase or decrease of derivative. In this problem, because the derivative function f'(x) is a quadratic function, in order to consider its positive and negative, we adopt the way of finding δ.

Knowledge involved

(1) quadratic function correlation (including quadratic inequality), you should be fine.

(2) Dependent on derivatives

When f'(x)≥0, f(x) increases monotonically.

When f'(x)≤0, f(x) decreases monotonically.

Please note that here, just like your underlined part, when f'(x)=0, it does not actually affect its monotonic increase. It can be considered that the derivative is actually the slope of the tangent of the point, in other words, it can be understood as a quantity of the rising or falling speed of the function, and a point can still maintain a continuous rising trend without changing.