What are the points of inflection, if any, of #f(x)=x^(1/3)e^(3x) #?

Question

What are the points of inflection, if any, of #f(x)=x^(1/3)e^(3x)   #?

Emilia Aguilar · Accepted Answer

there are two of them and exactly they are #x=\frac{-1-sqrt{3}}{9}# and #x=\frac{-1+sqrt{3}}{9}# It is matter of deriving twice #f(x)# #f'(x)=e^{3x}(\frac{1}{3root3{x^2}}+3root3{x})# that after a simple algebraic manipulation yelds #f'(x)=e^{3x}(\frac{1+9x}{3root3{x^2}})# so that the point of minimum has got abscissa #x=-\frac{1}{9}# By deriving #f'(x)# we obtain #f''(x)=\frac{-2}{9root3{x^5}}+\frac{2}{root3{x^2}}+9root3{x}# that after a simple manipulation turns to be #f''(x)=\frac{-2+18x+81x^2}{9root3{x^5}}# whose roots are just the abscissas of the inflection points

Scarlett Allison · Answer

undefined To find the points of inflection of $ f(x) = x^{1/3} e^{3x} $, we need to determine where the concavity changes.

First, calculate the second derivative of $ f(x) $ and set it equal to zero to find the potential points of inflection.

The first derivative of $ f(x) $ is:
$$ f'(x) = \frac{d}{dx} \left( x^{1/3} e^{3x} \right) $$

Using the product rule and the chain rule, we find:
$$ f'(x) = \frac{1}{3} x^{-2/3} e^{3x} + x^{1/3} \cdot 3 e^{3x} $$

Simplify this expression:
$$ f'(x) = \frac{1}{3x^{2/3}} e^{3x} + 3x^{1/3} e^{3x} $$

Now, differentiate $ f'(x) $ to find the second derivative:
$$ f''(x) = \frac{d}{dx} \left( \frac{1}{3x^{2/3}} e^{3x} + 3x^{1/3} e^{3x} \right) $$

Using the product rule and the chain rule again, we get:
$$ f''(x) = \frac{2}{9x^{5/3}} e^{3x} - \frac{1}{3x^{5/3}} e^{3x} + \frac{1}{x^{2/3}} \cdot 3 e^{3x} + 3x^{1/3} \cdot 3 e^{3x} $$

Simplify this expression:
$$ f''(x) = \frac{2}{9x^{5/3}} e^{3x} - \frac{1}{3x^{5/3}} e^{3x} + \frac{3}{x^{2/3}} e^{3x} + 3x^{1/3} \cdot 3 e^{3x} $$

Now, set $ f''(x) $ equal to zero and solve for $ x $ to find the potential points of inflection. Then, check the concavity around these points to confirm if they are points of inflection.

After solving $ f''(x) = 0 $, the points of inflection are obtained.