Sketch the curve and show intercepts, maxima and minima and points of inflection:
f(x)
5
x
-5
4
x

Solution

We define our function
In[]:=
f[x_]=x^5-5x^4
Out[]=
-5
4
x
+
5
x
In[]:=
FunctionDomain[f[x],x]
Out[]=
True
◼
  • We compute the x-intercepts
    • In[]:=
    Solve[f[x]==0,x]
    Out[]=
    {{x0},{x0},{x0},{x0},{x5}}
    The we calculate the y - intercepts
    In[]:=
    f[0]
    Out[]=
    0
    ◼
  • We find the asymptotes
    • In[]:=
    ResourceFunction["Asymptotes"][x^5-5x^4,x,y]
    Out[]=
    

    First Derivative

    ◼
  • Then we compute the first derivative of
    f(x)
    • In[]:=
    f'[x]
    Out[]=
    -20
    3
    x
    +5
    4
    x
    and then we calculate the critical points
    In[]:=
    Solve[f'[x]==0,x]
    Out[]=
    {{x0},{x0},{x0},{x4}}
    Thus, the monotonicity of the function is as follows
    In[]:=
    [◼]
    FunctionMonotonicity
    [f[x],x]
    Out[]=
    Increasingx≤0||4≤x,Decreasing0≤x≤4,ConstantFalse
    Hence,
    In[]:=
    f[0]
    Out[]=
    0
    is the max and,
    In[]:=
    f[4]
    Out[]=
    -256
    is the minimum .

    Second Derivative

    The second derivative can tell us the shape of a curve at any point .
    In[]:=
    f''[x]
    Out[]=
    -60
    2
    x
    +20
    3
    x
    We now use the second derivative to find points of inflection :
    In[]:=
    Solve[f''[x]==0,x]
    Out[]=
    {{x0},{x0},{x3}}
    In[]:=
    ResourceFunction["FunctionConcavity"][f[x],x]
    Out[]=
    ConcaveUpx>3,ConcaveDownx<0||0<x<3
    and
    In[]:=
    f[3]
    Out[]=
    -162
    So
    (3,-162)
    is the inflection point.
    ◼
  • Now, we are ready to sketch the curve :
    • In[]:=
    Plot[f[x],{x,-3,6},AxesLabel{"x","y"},PlotStyle->{Thick},Epilog->{Green,PointSize@Large,Point[{0,0}],Point[{5,0}],Red,PointSize@Large,Point[{3,-162}],Magenta,PointSize@Large,Point[{4,-256}]},PlotLabel->"Graph of
    y
    5
    x
    -5
    4
    x
    "]
    Out[]=
    The following points are indicated with dots:
    ​
    x-
    intercepts
    (0,0)
    and
    (5,0)
    (green dots)
    Point of inflection (
    3,-162)
    (red dot)
    Local maximum
    (0,0)
    (green dot)
    Local minimum
    (4,256)
    (magenta dot)