Pregunta
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© Find the critical points of . Sketch the graph of .

Ask by Hall Hills. in Kenya
Jan 23,2025

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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 :
Graph of f(x) = x^(1/3)(x - 4)
Key Features to Include in the Sketch:
  1. Critical Points:
    • Point at : This is where the derivative does not exist.
    • Point at : This is where the derivative is zero.
  2. 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.
  3. End Behavior:
    • As , behaves like , which increases to infinity.
    • As , behaves like but remains positive (since even roots yield positive values).
  4. Additional Points:
    • Plot a few more points for accuracy, such as (where ).
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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