Pregunta
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Respuesta verificada por el tutor
Responder
The critical points of
are at
and
. The graph of
has a critical point at
and another at
.
Solución
To find the critical points of the function
, we’ll follow these steps:
1. Find the Derivative
First, let’s differentiate
with respect to
:
Differentiating term by term:
2. Determine Where
or
Does Not Exist
a. Setting
:
Multiply both sides by
to eliminate the fractions and negative exponents:
b. Checking Where
Does Not Exist:
The derivative
involves the term
, which is undefined at
. Therefore,
is also a critical point.
3. Identify the Critical Points
-
At
: So, one critical point is. -
At
: Another critical point is.
4. Summary of Critical Points
5. Sketch of the Graph of
Here’s a rough sketch of the function
:

Key Features to Include in the Sketch:
-
Critical Points:
- Point at
: This is where the derivative does not exist. - Point at
: This is where the derivative is zero.
- Point at
-
Behavior Around Critical Points:
- At
: The function changes its slope direction, indicating a possible inflection point. - At
: This is likely a local minimum since the function decreases before and increases after.
- At
-
End Behavior:
- As
, behaves like , which increases to infinity. - As
, behaves like but remains positive (since even roots yield positive values).
- As
-
Additional Points:
- Plot a few more points for accuracy, such as
(where ).
- Plot a few more points for accuracy, such as
Note: The actual graph may require more detailed plotting for precision, especially around
due to the cube root function.
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Bonus Knowledge
To find the critical points of
, we first need to compute the derivative
. Using the product rule, we have:
Setting
gives us
, or
as a critical point. We also consider where the derivative is undefined; this occurs at
.
To sketch the graph of
, note that:
- At
, . - At
, . - At
, .
The function has the following intervals for behavior analysis:
- On
, will approach . - On
, decreases and touches zero at and reaches a negative value at . - On
, increases to touch the x-axis again at .
Thus, the graph will showcase a curve starting from the origin, dipping below the x-axis at
, and rising back to touch the axis at
.
Now you can visualize a connected and continuous graph cutting through the points identified to get an idea of how
behaves overall!

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