# Proof of No Local Max/Min

• Oct 9th 2008, 10:58 PM
banshee.beat
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\$
• Oct 9th 2008, 11:44 PM
Jhevon
Quote:

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