Critical Points and Points of Inflection

Question

Use derivatives to find x coordinates of all critical points of f prime and f double prime
1) f(x)= xe^-x
2) f(x)=-x^(5/3)+5x^(2/3) (for the first derivative I see where zero comes from but I don’t understand where the 2 comes from.

Answer 1

1) f(x) = xe^-x

f'(x) = e^-x - xe^-x = e^-x(1-x)

f"(x) = -e^-x - e^-x + xe^-x = -2e^-x + xe^-x = e^-x(x-2)

 

To get critical points, set f'(x) equal to zero.

f'(x) =  e^-x(1-x) = 0 

e^-x will never be zero so set 1-x = 0, you'd get x = 1

 

To get reflection points, set f"(x) equal to zero 

f"(x) = e^-x(x-2) = 0 

Again, e-x will never be zero so set x-2 = 0, you'd get x = 2

 

2) f(x) = x^5/3 + 5x^2/3

f'(x) = 5/3x^2/3 + 10/3x^-1/3 = 5/3x^-1/3(x+2)

f"(x) = 10/9x^-1/3 - 10/9x^-4/3 = 10/9x^-1/3(1-x⁴)

 

To get critical points, set f'(x) equal to zero

f'(x) = 5/3x^-1/3(x+2) = 0 

5/3x^-1/3 = 0 

x = 0 

x+2 = 0 

x = -2

 

Now, to get the reflection points, set f"(x) to zero: 

f"(x) = 10/9x^-1/3(1-x⁴) = 0 

10/9x-^1/3= 0 

x = 0 

1-x⁴ = 0 

x = -1,1