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Suppose that $ f $ and $ g $ are continuous on $ [a, b] $ and differentiable on $ (a, b) $. Suppose also that $ f(a) = g(a) $ and $ f'(x) \leqslant g'(x) $ for $ a < x < b $. Prove that $ f(b) < g(b) $. [Hint: Apply the Mean Value Theorem to the function $ h = f - g $.]
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Calculus 1 / AB
Calculus 2 / BC
Applications of Differentiation
The Mean Value Theorem
Harvey Mudd College
University of Nottingham
In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.
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Suppose that $f$ and $g$ a…
Prove the Mean Value Theor…
Let $f$ and $g$ be functio…
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