Looking at tangent lines

**Kipster1203** · April 16th 2010, 06:40 AM

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 $(c,f(c))$ .

**Drexel28** · April 16th 2010, 09:31 AM

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!

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