# What are the points of inflection, if any, of #f(x)=12x^3 -16x^2 +x +7 #?

Find where the second derivative is equal to 0.

Points of inflection in a function are those points where the function changes its concavity, between convex (i.e. concave up), and concave (i.e. concave down). Because a positive second derivative denotes a point where the function is convex and a negative one a point where the function is concave, at a point where concavity changes the second derivative must be 0.

Thus, step 1 is finding the second derivative using the Power Rule:

Now, we set this equal to 0 to find the appropriate x value or values. Since our second derivative is linear, we know there will be only one appropriate x value, and we know that the second derivative will change from positive to negative or negative to positive at that point. Thus, we know it will be an inflection point.

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To find the points of inflection of the function ( f(x) = 12x^3 - 16x^2 + x + 7 ), follow these steps:

- Compute the second derivative of the function, ( f''(x) ).
- Set ( f''(x) = 0 ) and solve for ( x ).
- Determine whether the concavity changes at each point where ( f''(x) = 0 ).
- If the concavity changes at a point, it is a point of inflection.

Let's start by finding the second derivative:

[ f(x) = 12x^3 - 16x^2 + x + 7 ]

[ f'(x) = 36x^2 - 32x + 1 ]

[ f''(x) = 72x - 32 ]

Now, set ( f''(x) = 0 ) and solve for ( x ):

[ 72x - 32 = 0 ]

[ x = \frac{32}{72} = \frac{4}{9} ]

Since the concavity changes at ( x = \frac{4}{9} ), it is a point of inflection.

Therefore, the point of inflection of the function ( f(x) = 12x^3 - 16x^2 + x + 7 ) is ( \left(\frac{4}{9}, f\left(\frac{4}{9}\right)\right) ).

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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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