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?

Theory and Examples

[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.

h(x) = |x + 2| − |x − 3|, −∞ < x < ∞

For the function graphed below, which interval contains a local maximum? $[-1,2]$, $[2,4]$, $[4,6]$, or $[6,8]$?

For the function $f\left(x\right)$ defined on the interval $[0,6]$, the graph shows a peak at $x=2$ and a trough at $x=5$. Which interval contains the local maximum of $f\left(x\right)$?

Over which interval is the graph of $f\left(x\right)=\frac{1}{2}{x}^2+5x+6$ increasing?

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

For the curve $y=1+40x^3-3x^5$, at which point does the tangent line have the largest slope?

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

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