How do you find all points of inflection given #y=-(4x+20)^(1/3)#?

Question

Isabella Ackerman · Accepted Answer

No points of inflection.

To find points of inflection, we find the second derivative of the function and set it equal to zero. We can then find the POI from the values that will make the equation true, i.e. equal to zero. #y=-(4x+20)^(1/3)# #=>y'=-4/3(4x+20)^(-2/3)# #=>y''=32/9(4x+20)^(-5/3)# We set the second derivative equal to zero... #32/9(4x+20)^(-5/3)=0# Which is equivalent to: #0=32/(9(4x+20)^(5/3))# And we can see that no matter what we do, we will never find a value of x that will make this true! Therefore, there are no points of inflection.

Samantha Adkison · Answer

undefined To find the points of inflection of the function $ y = -(4x + 20)^{\frac{1}{3}} $, follow these steps:

1. Find the second derivative of the function.
2. Set the second derivative equal to zero and solve for $ x $.
3. Determine the points where the concavity changes by examining the sign of the second derivative around the critical points found in step 2.

Let's proceed with finding the points of inflection:

1. Find the first derivative:
$$ y' = -\frac{1}{3}(4x + 20)^{-\frac{2}{3}} \cdot 4 = -\frac{4}{3}(4x + 20)^{-\frac{2}{3}} $$

2. Find the second derivative:
$$ y'' = \frac{8}{9}(4x + 20)^{-\frac{5}{3}} $$

3. Set the second derivative equal to zero and solve for $ x $:
$$ \frac{8}{9}(4x + 20)^{-\frac{5}{3}} = 0 $$
$$ \Rightarrow (4x + 20)^{-\frac{5}{3}} = 0 $$
$$ \Rightarrow \text{No real solutions.} $$

Since there are no solutions to set the second derivative equal to zero, we look for the points where the second derivative changes sign. The function $ y'' $ is always positive (since it's the square of a real number), which means the concavity doesn't change. Therefore, there are no points of inflection for the given function.