Functions $f$, $g$, and $h$ are twice-differentiable functions with $g(2) = h(2) = 4$. The line $y = 4 + \dfrac{2}{3}(x - 2)$ is tangent to both the graph of $g$ at $x = 2$ and the graph of $h$ at $x = 2$. (a) Find $h'(2)$. (b) Let $a$ be the function given by $a(x) = 3x^3 h(x)$. Write an expression for $a'(x)$. Find $a'(2)$. (c) The function $h$ satisfies $h(x) = \dfrac{x^2 - 4}{1 - (f(x))^3}$ for $x \neq 2$. It is known that $\lim_{x \to 2} h(x)$ can be evaluated using L'H\^{o}pital's Rule. Use $\lim_{x \to 2} h(x)$ to find $f(2)$ and $f'(2)$. Show the work that leads to your answers. (d) It is known that $g(x) \leq h(x)$ for $1 < x < 3$. Let $k$ be a function satisfying $g(x) \leq k(x) \leq h(x)$ for $1 < x < 3$. Is $k$ continuous at $x = 2$? Justify your answer.
Functions $f$, $g$, and $h$ are twice-differentiable functions with $g(2) = h(2) = 4$. The line $y = 4 + \dfrac{2}{3}(x - 2)$ is tangent to both the graph of $g$ at $x = 2$ and the graph of $h$ at $x = 2$.
(a) Find $h'(2)$.
(b) Let $a$ be the function given by $a(x) = 3x^3 h(x)$. Write an expression for $a'(x)$. Find $a'(2)$.
(c) The function $h$ satisfies $h(x) = \dfrac{x^2 - 4}{1 - (f(x))^3}$ for $x \neq 2$. It is known that $\lim_{x \to 2} h(x)$ can be evaluated using L'H\^{o}pital's Rule. Use $\lim_{x \to 2} h(x)$ to find $f(2)$ and $f'(2)$. Show the work that leads to your answers.
(d) It is known that $g(x) \leq h(x)$ for $1 < x < 3$. Let $k$ be a function satisfying $g(x) \leq k(x) \leq h(x)$ for $1 < x < 3$. Is $k$ continuous at $x = 2$? Justify your answer.