# How do you find the maximum, minimum and inflection points and concavity for the function #f(x)=6x^6+12x+6#?

Relative minimum at

Always concave up and no inflection points.

Now find the function's relative minimum and maximum using the sign test.

As for the function's concavity, we can conclude that it has no inflection points and is always concave upward because the second derivative is always positive.

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To find the maximum and minimum points:

- Take the first derivative of the function to find critical points.
- Set the first derivative equal to zero and solve for ( x ).
- Use the second derivative test to determine whether the critical points correspond to maximum or minimum points.

To find inflection points and concavity:

- Take the second derivative of the function to find points of inflection.
- Set the second derivative equal to zero and solve for ( x ).
- Determine the concavity of the function by analyzing the sign of the second derivative in intervals between critical points.

First Derivative: ( f'(x) = 36x^5 + 12 )

Set ( f'(x) ) equal to zero and solve for ( x ): ( 36x^5 + 12 = 0 ) ( x^5 = -\frac{1}{3} )

Second Derivative: ( f''(x) = 180x^4 )

Critical points: There are no real solutions for ( x^5 = -\frac{1}{3} ), so there are no critical points.

Concavity: Since there are no critical points, we consider the concavity of the function for all ( x ).

Since ( f''(x) = 180x^4 ) is always positive for real values of ( x ), the function is concave up for all ( x ).

There are no maximum or minimum points, and the function is concave up for all ( x ). Hence, there are no inflection points.

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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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