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.

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 \(a+b\)?

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\(f(x)\) 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 \(f(15)\)?

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\(f(x)\) 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 \(a+b\)?

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