Question

What are the points of inflection, if any, of `f(x)= -14x^3 + 19x^2 - x - 2`?

Answer 1

Points of inflection are `(0.64,1.47)` and `(-0.21, -0.812)`

Explanation:

To find points of inflection, differentiate the expression to get the gradients, and then find the points where the gradient is zero.
`f(x) = -14x^3 +19x^2 - x - 2`

`f'(x)=-42x^2 +38x -1`

for points of inflection, `-42x^2+38x-1 = 0`

`42x^2 -38x +1 = 0`

This does not factorise so use the quadratic equation `x = (-b+-sqrt(b^2-4ac))/(2a)`

`x=(38+-sqrt(1444-168))/84`

`x~~(38+-35.72)/84`

`x~~0.64` or `x~~-0.21`

Substitute back into the original expression to find the values of `f(x)`

Points of inflection are `(0.64,1.47)` and `(-0.21, -0.812)`