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

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.

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

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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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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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