# Thread: Proof of No Local Max/Min

1. ## Proof of No Local Max/Min

Okay.. so I've been doing calculus for about 4 hours at this point. And I haven't the slightest clue how to go about this proof... and my brain feels like jello.

Prove that the function

$\displaystyle f(x)=x^{101}+x^{51}+x+1$

has neither a local maximum nor a local minimum.

I took the derivative and now do not have any clue of what I need to do... how can I find critical points for it? Please help.

$\displaystyle f'(x)=101x^{100}+51x^{50}+1$

2. Originally Posted by banshee.beat
Okay.. so I've been doing calculus for about 4 hours at this point. And I haven't the slightest clue how to go about this proof... and my brain feels like jello.

Prove that the function

$\displaystyle f(x)=x^{101}+x^{51}+x+1$

has neither a local maximum nor a local minimum.

I took the derivative and now do not have any clue of what I need to do... how can I find critical points for it? Please help.

$\displaystyle f'(x)=101x^{100}+51x^{50}+1$
notice that this is actually a quadratic in disguise. it is quadratic in $\displaystyle x^{50}$. if you can't see it, let $\displaystyle a = x^{50}$, then you have $\displaystyle 101a^2 + 51a + 1$

now can you find the critical points? set the function to zero and see what happens