What are the points of inflection of #f(x)=12x^3 + 3x^2 + 42x #?

Answer 1

#x=-1/12#

Points of inflection are correlated with shifts in the concavity of the function and occur when the second derivative of a function changes sign, either from positive to negative or vice versa.

Find the function's second derivative first.

#f(x)=12x^3+3x^2+42x#
#f'(x)=36x^2+6x+42#
#f''(x)=72x+6#
The sign of #f''(x)# could shift when #f''(x)=0#, so set #72x+6=0#.
#72x+6=0#
#x=-6/72#
#x=-1/12#
Analyze the sign of #f''(x)# around the possible point of inflection #x=-1/12#.
When #mathbf(x<-1/12)#:
#f''(-1)=-72+6=-66#
This is #<0#.
When #mathbf(x> -1/12)#:
#f''(0)=6#
This is #>0#.
Thus, the sign of #f''(x)# does change around the point when #x=-1/12#.
We can check a graph of #f(x)#:

graph{[-2.5, 2.5, -200, 200]} 12x^3+3x^2+42x

The concavity does seem to shift very close to #x=0#.
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Answer 2

To find the points of inflection of the function ( f(x) = 12x^3 + 3x^2 + 42x ), we need to find where the concavity changes.

The second derivative of ( f(x) ), denoted as ( f''(x) ), gives us information about concavity.

The second derivative of ( f(x) = 12x^3 + 3x^2 + 42x ) is ( f''(x) = 72x + 6 ).

To find the points of inflection, we set ( f''(x) = 0 ) and solve for ( x ).

( 72x + 6 = 0 )

Solving for ( x ), we get ( x = -\frac{1}{12} ).

So, the point of inflection is ( (-\frac{1}{12}, f(-\frac{1}{12})) ).

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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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