I don't understand why does the $\theta (1-\theta)(f(y)-f(x))(g(x)-g(y)) \le 0 $ means both f and g are increasing or decreasing,can anyone take an example?
By the way,the solution said that $\theta (1-\theta)(f(y)-f(x))(g(x)-g(y)) \le 0 $ means both f and g are increasing or decreasing,however,in the last formula,it just let $\theta (1-\theta)(f(y)-f(x))(g(x)-g(y)) = 0 $,why? Is't it $\le 0$,not just $=0$ ?
Your interpretation of the logic is not quite right. It is saying that IF $f$ and $g$ are both increasing or both decreasing, THEN the third term is non-positive. Your interpretation is the converse statement: IF the third term is non-positive, THEN $f$ and $g$ are both increasing or both decreasing. In general, a statement is not logically equivalent to its converse.
Notice then that the correct interpretation certainly holds: If $f$ and $g$ are both increasing or both dcreasing, then $f(y)-f(x)$ and $g(x)-g(y)$ will have different signs, and since $0 \le \theta \le 1$, we have $\theta(1-\theta) \ge 0$. Hence $$\theta(1-\theta)(f(y)-f(x))(g(x)-g(y)) \le 0.$$
As for the final conclusion, I'm going to assume that we are given information in the problem statement that allows us to say $f$ and $g$ are both increasing or both decreasing. Then by the previous remarks, we have an inequality of the form $$f(\theta x + (1-\theta)y)g(\theta x + (1-\theta)y) \le A + B + C,$$ where $C \le 0$. In the last step we aren't claiming $C = 0$, we are simply observing that adding nothing gives a bigger number than adding a non-positive number, so if we throw out the $C$ we still have an inequality.