How do you sketch the curve #y=x^3+6x^2+9x# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
See graph and explanation
Polynomial graphs have no asymptotes.
As x to +oo, y = x^3(1+3/x)^2 to +oo, showing end behavior of
So, there are no global extrema.
Turning points or points of inflexion at (1, 4) and (3, 0)
This POI is marked, in the graph.
graph{(x(x+3)^2y)((x+2)^2+(y+2)^2.01)=0 [10, 10, 10, 5]}
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To sketch the curve ( y = x^3 + 6x^2 + 9x ), follow these steps:

Find Critical Points: Set the derivative equal to zero and solve for ( x ) to find critical points.

Determine Sign of Derivative: Use the first derivative test to determine the intervals where the function is increasing or decreasing.

Find Inflection Points: Set the second derivative equal to zero and solve for ( x ) to find inflection points.

Determine Concavity: Use the second derivative test to determine the intervals where the function is concave up or concave down.

Find Asymptotes: Determine any vertical, horizontal, or slant asymptotes by analyzing the behavior of the function as ( x ) approaches infinity or negative infinity.

Find Intercepts: Find the ( y )intercept by evaluating the function at ( x = 0 ). Find the ( x )intercepts by setting ( y ) equal to zero and solving for ( x ).
Once you have gathered this information, plot the points, sketch the curve connecting them, and label any relevant features such as maximums, minimums, inflection points, and asymptotes.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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