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Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.

Mean Value Theorem


For function \(f(x)=x^3+9\), constant \(\theta\) satisfies \[f(x+h)-f(x)=hf'(x+\theta h),\] where \(x > 0\), \(h > 0\) and \(0 < \theta < 1\). If the value of \(\displaystyle \lim_{h \to 0} \theta\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is \(a+b\)?

\(f(x)\) is a function that is continuous and differentiable in the domain \(\left[7, 15 \right]\). If \(f(7) = 21\) and \(f'(x) \leq 14\) for all \(7 \leq x \leq 15\), what is the maximum possible value of \(f(15)\)?

\(f(x)\) is a differentiable function that satisfies \(5 \leq f'(x) \leq 14\) for all \(x\). Let \(a\) and \(b\) be the maximum and minimum values, respectively, that \(f(11)-f(3)\) can possibly have, then what is the value of \(a+b\)?

For the function \(f(x) = x^2\), what is the value of \(x\) in the interval \([0,\ 44]\) that satisfies the Mean Value Theorem?

For \(f(x) = \sqrt{x-1}\), what is the value of \(x\) in the interval \([1,\ 38]\) that satisfies the mean value theorem?


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