Proof of No Local Max/Min

**banshee.beat** · October 9th 2008, 10:58 PM

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

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

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.

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

**Jhevon** · October 9th 2008, 11:44 PM

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

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

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.

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

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

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

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