Pretty much as in the title, except one major condition: g(x_0) = m*x_0 + b.(adsbygoogle = window.adsbygoogle || []).push({});

So, conditions are:

g defined on the reals and is differentiable at x_0.

g(x_0) = m*x_0 + b

mx + b <= g(x) for all x

Then show m = g'(x_0)

I would love to use the intermediate value theorem, or extreme value theorem, or mean value theorem, but since we can only say g is differentiable at x_0 those can't apply since we don't have any intervals where those theorems would apply. Just wondering if there's an obvious theorem to apply here. Thanks!

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# If g is diff'able at x_0, g(x) <= mx + b, show g'(x_0) = m

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