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Answer
The function
Solution
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Identify the function and its domainThe function given isThe domain of a polynomial function is all real numbers, so
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Intercepts
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Y-intercept: Set
: Thus, the y-intercept is at. -
X-intercept: Set
: Thus, the x-intercept is also at.
-
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SymmetryTo check for symmetry, substitute
for : Since, the function is \emph{odd} and symmetric with respect to the origin. -
First Derivative (Rate of Change/Increasing or Decreasing)Compute the first derivative:
- Since
for all (with equality only when ), we have . - Therefore, the function is increasing for all
(strictly increasing except possibly at the stationary point where the derivative is ).
- Since
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Second Derivative (Concavity and Inflection Point)Compute the second derivative:
- For
: so the function is concave upward. - For
: so the function is concave downward. - The concavity changes at
, so there is an inflection point at .
- For
-
Summary
- Function:
- Domain:
- Intercept: Both
and intercept at - Symmetry: Odd function (symmetric with respect to the origin)
- Increasing/Decreasing: Always increasing (
) - Concavity:
- Concave downward for
( ) - Concave upward for
( )
- Concave downward for
- Inflection Point:
where the concavity changes.
- Function:
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Did you know that the equation
When working with cubic functions, a common mistake is to forget about the critical points and inflection points! Make sure to differentiate the function to find where the slope is zero (which corresponds to local maxima and minima) and where the concavity changes. Analyzing these points can help sketch the graph accurately and understand its behavior better!
