Calculus

Differentiability

Mean Value Theorem

         

For function f(x)=x3+9f(x)=x^3+9, constant θ\theta satisfies f(x+h)f(x)=hf(x+θh),f(x+h)-f(x)=hf'(x+\theta h), where x>0x > 0, h>0h > 0 and 0<θ<10 < \theta < 1. If the value of limh0θ\displaystyle \lim_{h \to 0} \theta can be expressed as ab\frac{a}{b}, where aa and bb are coprime positive integers, what is a+ba+b?

f(x)f(x) is a function that is continuous and differentiable in the domain [7,15]\left[7, 15 \right]. If f(7)=21f(7) = 21 and f(x)14f'(x) \leq 14 for all 7x157 \leq x \leq 15, what is the maximum possible value of f(15)?f(15)?

f(x)f(x) is a differentiable function that satisfies 5f(x)145 \leq f'(x) \leq 14 for all xx. Let aa and bb be the maximum and minimum values, respectively, that f(11)f(3)f(11)-f(3) can possibly have, then what is the value of a+b?a+b?

For the function f(x)=x2f(x) = x^2, what is the value of xx in the interval [0, 44][0,\ 44] that satisfies the Mean Value Theorem?

For f(x)=x1f(x) = \sqrt{x-1}, what is the value of xx in the interval [1, 38][1,\ 38] that satisfies the mean value theorem?

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