5. Graphical Applications of Derivatives

Consider the following graph of $f\left(x\right)$. Which of the following points are inflection points of $f$?

{Use of Tech} Critical points and extreme values

a. Find the critical points of the following functions on the given interval. Use a root finder, if necessary.

b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.

c. Find the absolute maximum and minimum values on the given interval, if they exist

h(x) (5-x)/(x² + 2x - 3) on [-10,10]

Suppose the graph of $f\left(x\right)$ is shown below. At which intervals is $f^{\prime }\left(x\right)$ $<0$?

For which values of $x$ is the function $f\left(x\right)=(x^2-3)e^{-x}$ increasing?

Theory and Examples

Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Find the exact length of the curve $y=2\surd 3x^{\frac{3}{2}}$ for $0\le x\le 6$.

For the function $f(x,y)=x^2+y^2-4x-6y+13$, which ordered pair is closest to a local minimum of the function?

Find the exact length of the curve $y=\ln (\sec (x\left)\right)$ for $0\le x\le \frac{\pi }{6}$.

{Use of Tech} Absolute maxima and minima

a. Find the critical points of f on the given interval.

b. Determine the absolute extreme values of f on the given interval.

c. Use a graphing utility to confirm your conclusions.

f(x) = 2ᶻ sin x on [-2,6]

Consider the function $q\left(x\right)=54x^4+125x$. What are the critical numbers of $q\left(x\right)$?

Let $f\left(x\right)=x^3-3x^2+2$. On which of the following intervals is $f\left(x\right)$ decreasing?

Suppose a continuous function $f\left(x\right)$ is defined on the closed interval $[a,b]$, and $x=c$ is a critical point in $(a,b)$. Which of the following is true about the curve at the point $x=c$?

Let $v\left(t\right)=t^3-6t^2+9t+2$ for $t$ in the interval $[0,5]$. At which value(s) of $t$ does $v\left(t\right)$ attain a local minimum?

Find the exact length of the curve $y=\frac{x^3}{3}+\frac{1}{4x}$ for $1\le x\le 2$.

Find the critical points of the given function.

$f(t)=\(\frac{6t}{t^2+1}\)$