11-9 Critical Points Worksheet

Let h(x)=((10(((x-2))^(2/3)))/x). A computer algebra system gives the derivative as h'(x)=((10(6-x))/(3(((x^2)(x-2))^(1/3)))).

What are the intervals of increase and the intervals of decrease?

What is the domain for the original function h(x)? What is the domain for the derivative h'(x)?

Domain of h(x):

Domain of h'(x):

Find all the critical numbers of h(x), and for each critical number determine if it is a local maximum, a local minimum, or neither.

Plot the graph in Desmos or a graphing calculator. Sketch the results below. Does your results in 1.1, 1.2 and 1.3 agree with the graph?

Graphics[Table[{Line[{{n,-10},{n,10}}],Line[{{-10,n},{10,n}}]},{n,-10,10,5}],PlotRange->{{-10,10},{-10,10}},GridLines->{Range[-10,10],Range[-10,10]},Axes->True,AxesStyle->Thick,PlotLabel->HoldForm[y=h[x]],BaseStyle->FontSize->14]

Let f(x)=((((x-1))^(1/3))/(x+2)). A computer algebra system gives the derivative as f'(x)=((5-2x)/(3(((x+2))^2)(((x-1))^(2/3)))).

What are the intervals of increase and the intervals of decrease?

What is the domain for the original function f(x)? What is the domain for the derivative f'(x)?

Domain of f(x):

Domain of f'(x):

Find all the critical numbers of f(x), and for each critical number determine if it is a local maximum, a local minimum, or neither. (Check your results with a graph from Desmos or a graphing calculator. Do they agree?)

Function g(x) has domain (-∞,∞). The graph shown below is of the derivative g'(x) (NOT g(x)). However, use the graph to answer questions about the original function g(x).

Show[Plot[0.5(((x-2.5))^3)+1.5(((x-2.5))^2)-2,{x,-1,5}],Plot[-x+3,{x,-5,-1}],Graphics[{PointSize[0.028],Point[{x,0.5(((x-2.5))^3)+1.5(((x-2.5))^2)-2}/.x->-1],Point[{-1,4}],White,PointSize[0.022],Point[{x,0.5(((x-2.5))^3)+1.5(((x-2.5))^2)-2}/.x->-1],Point[{-1,4}]}],PlotRange->{{-6,6},{-6,6}},AspectRatio->Automatic,GridLines->{Range[-6,6],Range[-6,6]},PlotLabel->HoldForm[y=g'[x]],BaseStyle->FontSize->14]

What are the intervals of increase for g(x). (What does that mean for g'(x)?)

What are the intervals of decrease for g(x). (What does that mean for g'(x)?)

Find all the critical points for g(x), and for each one determine if it is a local maximum, a local minimum, or neither.