# Thread: Looking at tangent lines

1. ## Looking at tangent lines

Suppose that $I\subset \mathbb{R}$ is an open interval and that $f''(x) \geq 0$ for all $x\in I$. If $c \in I$ , show that the part of the graph of $f$ on $I$ is never below the tangent line to the graph at $(c,f(c))$ .

2. Originally Posted by Kipster1203
Suppose that $I\subset \mathbb{R}$ is an open interval and that $f''(x) \geq 0$ for all $x\in I$. If $c \in I$ , show that the part of the graph of $f$ on $I$ is never below the tangent line to the graph at $(c,f(c))$ .
So, let $\varphi(x)=f(x)-f'(c)(x-c)-f(c)$. Then, $\varphi(x)$ is the difference between $f$ and the tangent line at $x=c$. So, what we want to do is show that $\varphi'(x)\leqslant 0,\text{ }x\leqslant c$ and $\varphi'(x)\geqslant 0,\text{ }x\geqslant c$. But, $\varphi'(x)=f'(x)-f'(c)$ and $\varphi''(x)$ and so if $x\leqslant c$ then $\varphi'(x)\leqslant \varphi'(c)=0$ and if $x\geqslant c$ then $\varphi'(x)\geqslant \varphi'(c)=0$. From prior comment the conclusion follows.

NOTE! The above is purposefully full of holes! Fill them in!