Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h^{\prime}(x) = \dfrac{x^{2} - 2}{x}$ for all $x \neq 0$. (a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer. (c) Write an equation for the line tangent to the graph of $h$ at $x = 4$. (d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?
Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h^{\prime}(x) = \dfrac{x^{2} - 2}{x}$ for all $x \neq 0$.\\
(a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers.\\
(b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer.\\
(c) Write an equation for the line tangent to the graph of $h$ at $x = 4$.\\
(d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?