how many real roots does 2x^5 +8x -7 have?

Anyone have any fancy ways to solove this?

I looked for stationary points by setting the derrivative to 0, to get 10x^4+8=0 so x^4=-4/5

Since x^4>0 there are no real stationary points so there must only be one real root?

It seems like it works on this problem, but only because there were no stationary points? Could they ask a problem with stationary points? Would you just have to investigate between those points?

Thanks

## 0568 #17

For Example: If the function only approached the x axis instead of going below it.

Since, the degree is 5 - ODDhow many real roots does 2x^5 +8x -7 have?

Anyone have any fancy ways to solove this?

I looked for stationary points by setting the derrivative to 0, to get 10x^4+8=0 so x^4=-4/5

Since x^4>0 there are no real stationary points so there must only be one real root?

It seems like it works on this problem, but only because there were no stationary points? Could they ask a problem with stationary points? Would you just have to investigate between those points?

Thanks

so f(x)------->+infinity when x----->+infinity

and f(x)------->-infinity when x----->-infinity

so we can find M, N st f(M)f(N)<0 so f(x)=0 has at least one root

Moreover, f'(x)=4x^2+8 >0 then f(x) has only one root