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Thread: Looking at tangent lines

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    Looking at tangent lines

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


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