26. Find all points of inflection.
f(x) = (5x-3)^(1/3)
Note: I have gotten
f''(x) = (-50/9)(5x-3)^(-5/3)
I would like to know how to figure out the zeros.
you want to have (-50/9)(5x-3)^(-5/3) = 0
then (5x-3)^(-5/3) = 0
or
1/(5x-3)^(5/3) = 0
Clearly this has no solution.
Think of it this way:
1 divided by what number would give you zero?? There is no such number
So since your equation has no solution, there is no point of inflection.
To find the points of inflection, we need to find the x-values where the second derivative, f''(x), equals zero or does not exist. In this case, f''(x) = (-50/9)(5x-3)^(-5/3).
To figure out the zeros of f''(x), we need to solve the equation (-50/9)(5x-3)^(-5/3) = 0.
To do this, we first notice that the factor (-50/9) is not equal to zero, which means we can exclude it when finding the zeros. Then the equation becomes (5x-3)^(-5/3) = 0.
Now, a non-zero real number raised to the power of -5/3 will never equal zero. Therefore, there are no x-values that make the second derivative zero.
This means that there are no points of inflection for the given function, f(x) = (5x-3)^(1/3).