How do you determine whether the function # f(x) = 2x^3 + 3x^2 - 432x # is concave up or concave down and its intervals?

Answer 1

f(x) is concave down on #(-oo;-1/2)# and concave up on #(-1/2;oo)#

You have to investigate the sign of the second derivative around its zero points (inflection points).

In this case,

#f'(x)=6x^2+6x-432# #f''(x)=12x+6#
#therefore f''(x)=0iffx=-1/2#
#therefore AAx<-1/2=>f''(x)<0# and hence the concavity is down.
#therefore AAx >-1/2=>f''(x)>0# and hence the concavity is up.
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Answer 2

To determine whether the function ( f(x) = 2x^3 + 3x^2 - 432x ) is concave up or concave down and its intervals, you need to find the second derivative of the function.

First, find the first derivative of ( f(x) ), then find the second derivative.

Then, examine the sign of the second derivative:

  • If the second derivative is positive for a certain interval, the function is concave up on that interval.
  • If the second derivative is negative for a certain interval, the function is concave down on that interval.

The intervals where the concavity changes (i.e., where the second derivative changes sign) are the points of inflection for the function.

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Answer from HIX Tutor

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.

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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