91Ó°ÊÓ

Determine the open intervals on which the graph is concave upward or concave downward. $$ y=-x^{3}+3 x^{2}-2 $$

Identify the open intervals on which the function is increasing or decreasing. $$f(x)=x^{2}-6 x+8$$

Find the value of the derivative (if it exists) at each indicated extremum. $$f(x)=\frac{x^{2}}{x^{2}+4}$$

Determine the open intervals on which the graph is concave upward or concave downward. $$ f(x)=\frac{24}{x^{2}+12} $$

Complete two iterations of Newton's Method for the function using the given initial guess. $$ f(x)=\sin x, \quad x_{1}=3 $$

Let \(f(x)=\frac{c}{x}+x^{2} .\) Determine all values of the constant \(c\) such that \(f\) has a relative minimum, but no relative maximum.

(a) Let \(f(x)=a x^{2}+b x+c, a \neq 0\), be a quadratic polynomial. How many points of inflection does the graph of \(f\) have? (b) Let \(f(x)=a x^{3}+b x^{2}+c x+d, a \neq 0\), be a cubic polynomial. How many points of inflection does the graph of \(f\) have? (c) Suppose the function \(y=f(x)\) satisfies the equation \(\frac{d y}{d x}=k y\left(1-\frac{y}{L}\right)\) where \(k\) and \(L\) are positive constants. Show that the graph of \(f\) has a point of inflection at the point where \(y=\frac{L}{2}\). (This equation is called the logistic differential equation.)

Identify the open intervals on which the function is increasing or decreasing. $$y=-(x+1)^{2}$$

Complete two iterations of Newton's Method for the function using the given initial guess. $$ f(x)=\tan x, \quad x_{1}=0.1 $$

Find the value of the derivative (if it exists) at each indicated extremum. $$f(x)=\cos \frac{\pi x}{2}$$