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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