How do you find all points of inflection given #y=x^5-2x^3#?

Question

Paisley Johnson · Accepted Answer

There is just one point of inflection at the origin

# y = x^5 - 2x^3 #

Differentiating wrt #x# we have'

# dy/dx = 5x^4 - 6x^2 #

At a critical point , # dy/dx = 0 #
# dy/dx = 0 => 5x^4 - 6x^2 = 0 #
# :. x^2(5x^2 - 6) = 0 #
So, Either # x^2 = 0 => x= 0#, or # 5x^2 - 6 = 0 => x =+- sqrt(6/5) #

So critical points occurs when #x=0, x =+- sqrt(6/5)#

Next we can find the nature of th critical points by looking at the second derivative:

# dy/dx = 5x^4 - 6x^2 #
# :. (d^2y)/(dx^2) = 20x^3 - 12x #
# :. (d^2y)/(dx^2) = 4x(5x^2 - 3) #

When # { (x=-sqrt(6/5), => (d^2y)/(dx^2)<0,=>,"maximum"), (x=0, => (d^2y)/(dx^2)=0,=>,"inflection"), (x =sqrt(6/5), => (d^2y)/(dx^2)>0,=>,"minimum") :} #

So therei is one point of inflection when #x=0 => y=0#

Ezra Adams · Answer

undefined To find all points of inflection for the function $ y = x^5 - 2x^3 $, follow these steps:

1. Find the second derivative of the function $ y $ with respect to $ x $.
2. Set the second derivative equal to zero and solve for $ x $ to find any potential points of inflection.
3. Test the concavity of the function on both sides of each potential point of inflection to confirm if they are indeed points of inflection.

Let's go through these steps:

1. Find the first derivative:
$$ \frac{dy}{dx} = 5x^4 - 6x^2 $$

2. Find the second derivative:
$$ \frac{d^2y}{dx^2} = \frac{d}{dx}(5x^4 - 6x^2) = 20x^3 - 12x $$

3. Set the second derivative equal to zero and solve for $ x $:
$$ 20x^3 - 12x = 0 $$
$$ 4x(5x^2 - 3) = 0 $$
$$ x = 0, \sqrt{\frac{3}{5}}, -\sqrt{\frac{3}{5}} $$

4. Test the concavity of the function around each potential point of inflection:
   - For $ x = 0 $: Test the signs of the second derivative on either side of $ x = 0 $. You'll find a change in concavity, indicating a point of inflection.
   - For $ x = \sqrt{\frac{3}{5}} $ and $ x = -\sqrt{\frac{3}{5}} $: Similarly, test the signs of the second derivative on either side of these points. You'll also find changes in concavity, indicating points of inflection.

So, the points of inflection for the function $ y = x^5 - 2x^3 $ are $ (0, 0) $, $ \left(\sqrt{\frac{3}{5}}, \sqrt{\frac{3^5}{5^5}} - 2\left(\frac{3}{5}\right)^{3/2}\right) $, and $ \left(-\sqrt{\frac{3}{5}}, \sqrt{\frac{3^5}{5^5}} - 2\left(\frac{3}{5}\right)^{3/2}\right) $.