Calculus

# 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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