# How do you sketch the graph #y=x^4-2x^3+2x# using the first and second derivatives?

See process below

1.- Domain is

2.- Analyze the roots of equation

Notice that

3.- First derivative

Analyze the sign of derivative.

Then function is increasing in

Derivative is zero in

A sketch is presented below

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To sketch the graph of ( y = x^4 - 2x^3 + 2x ) using the first and second derivatives:

- Find the first derivative of ( y ) with respect to ( x ) and locate the critical points.
- Find the second derivative of ( y ) with respect to ( x ) and determine the concavity of the function.
- Locate the inflection points by setting the second derivative equal to zero and solving for ( x ).
- Plot the critical points, inflection points, and any intercepts of the function.
- Sketch the graph, indicating increasing/decreasing intervals based on the first derivative and concave up/down intervals based on the second derivative.

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When evaluating a one-sided 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 one-sided 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 one-sided 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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- Find y' and y''? #y = x^2ln(2x)#

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