1. ## Minima problem

THe number of minima of the polynomial 10 x^6 - 24 x^5 + 15 x^4 + 40 x^2 + 108 is ?

(I took derivative and found a polynomial of degree 5. But making its roots and then testing each one for minima is going to be reaaaly long. Is there some easier way to do this question? )

Thanks

2. Originally Posted by champrock
THe number of minima of the polynomial 10 x^6 - 24 x^5 + 15 x^4 + 40 x^2 + 108 is ?

(I took derivative and found a polynomial of degree 5. But making its roots and then testing each one for minima is going to be reaaaly long. Is there some easier way to do this question? )

Thanks
Do you know the second derivative test?

If, at a stationary point $\displaystyle x = a, f''(a)>0$, then $\displaystyle a$ is a minimum.

So

$\displaystyle f'(x) = 60x^5 - 120x^4 + 60x^3 + 80x$.

Find the stationary points...

$\displaystyle 0 = 60x^5 - 120x^4 + 60x^3 + 80x$

$\displaystyle 0 = 20x(3x^4 - 6x^3 + 4)$

I'll test one of the stationary points for you... Clearly $\displaystyle x= 0$ is a stationary point.

So let's take the second derivative, and see what it's sign is when $\displaystyle x = 0$.

$\displaystyle f''(x) = 300x^4 - 480x^3 + 180x^2 + 80$

$\displaystyle f''(0) = 300(0)^4 - 480(0)^3 + 180(0)^2 + 80$

$\displaystyle f''(0) = 80 > 0$

Therefore $\displaystyle x = 0$ is a minimum.

Test any other stationary points in the same manner.