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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