Question

How many inflection points are in the graph of `f(x)= (x^7)/42 - (3x^6)/10 + (6x^5)/5 - (4x^4)/3`?

Answer 1

There is one inflection point.

Explanation:

`f(x)= (x^7)/42 - (3x^6)/10 + (6x^5)/5 - (4x^4)/3`

`f'(x)= (x^6)/6 - (9x^5)/5 + 6x^4 - (16x^3)/3`

`f''(x)= x^5 - 9x^4 + 24x^3 - 16x^2`

Finding the zeros of `f''`

`f''(x)= x^5 - 9x^4 + 24x^3 - 16x^2`

`= x^2(x^3-9x^2+24x-16)`

Rational Zeros Theorem and checking (or inspection of likely candidates or adding the coefficients) we see that `1` is a zero. Therefore `x-1` is a factor.
Factor or do the division to get:

`f''(x) = x^2(x-1)(x^2-8x+16)`

`= x^2(x-1)(x-4)^2`

The zeros are `0, 1, "and " 4`.

The sign of `f''(x)` changes only at the single zero `x=1`

Therefore the only inflection point is `(1, f(1))`.