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Math Help - Looking at tangent lines

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    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)) .
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Kipster1203 View Post
    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.


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