RooftopTheorem

# RooftopTheorem - Rooftop Theorem for Concave functions This...

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Rooftop Theorem for Concave functions This theorem asserts that if f is a differentiable concave function of a single variable, then at any point x in the domain of f , the tangent line through the point ( x, f ( x )) lies entirely above the graph of f . You should draw a picture. Theorem 1. If f is a continuously differentiable concave function of a single variable, defined on a real interval I , then for all x 1 and x 2 in I , f ( x 1 ) + ( x 2 - x 1 ) f 0 ( x 1 ) f ( x 2 ) . Geometrically, this theorem says that the tangent line to the graph of f passing through any point ( x 1 , f ( x 1 )) must lie entirely on or above the graph of f . You should draw a couple of pictures to convince yourself of this geometry. Proof. Since f is a concave function, it must be that for all x 1 and x 2 in I , and all t [0 , 1], f ((1 - t ) x 1 + tx 2 ) (1 - t ) f ( x 1 ) + tf ( x 2 ) . (1) Rearranging terms, we see that Equation 1 is equivalent to f ( x 1 + t ( x 2 - x 1 )) - f ( x 1 ) t ( f ( x 2 ) - f ( x 1 )) . (2) Dividing both sides of equation 2 by t , we have f ( x 1 + t ( x 2 - x 1 )) - f ( x 1 ) t f ( x 2 ) - f ( x 1 ) (3)

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