# How do you find the inflection points of the graph of the function: f(x)= (x^7/42) - (3x^6/10) + (6x^5/5) - (4x^4/3)?

Oct 18, 2015

See the explanation.

#### Explanation:

For $f \left(x\right) = \left({x}^{7} / 42\right) - \left(3 {x}^{6} / 10\right) + \left(6 {x}^{5} / 5\right) - \left(4 {x}^{4} / 3\right)$, we get

$f ' ' \left(x\right) = {x}^{5} - 9 {x}^{4} + 24 {x}^{3} - 16 {x}^{2}$

$= {x}^{2} \left({x}^{3} - 9 {x}^{2} + 24 x - 16\right)$

Looking for rational zeros of the cubic, we note that $1$ is a zro, so $x - 1$ is a factor. Dividing by $x - 1$ gets us:

$= {x}^{2} \left(x - 1\right) \left({x}^{2} - 8 x + 16\right)$

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

The zeros of $f ' '$ are $0$, $1$, and $4$.

The only one at which $f ' '$ changes sign is $1$.

The only inflection point is $\left(1 , f \left(1\right)\right)$

(I'll leave it to the student to find $f \left(1\right)$)