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

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Question

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

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Isaiah Allen · Accepted Answer

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: #f(x) = 12x^3 - 16x^2 + x + 7 -> (df)/dx = 36x^2-32x+1 -> (d^2f)/dx^2 = 72x - 32# 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. #(d^2f)/dx^2 = 72x-32 = 0 -> 72x = 32 -> x = 32/72 = 16/36 = 8/18 = 4/9# The only inflection point is at #x=4/9#, and the y coordinate is... #f(4/9) = 12(4/9)^3 -16(4/9)^2 + 4/9 + 7 = 12(64/729)-16(16/81) + 4/9 + 7 ~5.337# The only point of inflection is at #(4/9, 5.337)#

Christopher Agresta · Answer

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

1. Compute the second derivative of the function, $ f''(x) $.
2. Set $ f''(x) = 0 $ and solve for $ x $.
3. Determine whether the concavity changes at each point where $ f''(x) = 0 $.
4. 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) $.