메모

증가함수의 정의 (1)

$$x_1<x_2⟹f(x_1)<f(x_2)$$

도함수가 양수이면 원함수는 증가함수이고 음수이면 원함수가 감소함수인가?

$$\begin{array}{rl}& \forall x\in \mathbb{R}\\ & x_1<x_2\\ & \frac{f(x_2)-f(x_1)}{x_2-x_1}=f^{\prime }(c)c\in (x_1,x_2)\\ & f^{\prime }(c)>0⟹f(x_2)-f(x_1)>0(\because x_1<x_2)\\ & \therefore f^{\prime }(x)>0⟹f(x_1)<f(x_2)\end{array}$$

fixed value theorem squeeze theorem rolle’s theorem mean value theorem