Proof: f(x) Has No Local Max/Min

[BrownianMan]

Show that the function f(x) = x^21 + x^11 + 13x does not have a local maximum or minimum.

So f '(x) = 21x^20 + 11x^10 + 13.

My reasoning is as follows:

Since the exponents (10 and 20) are even, 21x^20 and 11x^10 can never be negative, and thus, summing them can never produce a negative number to make the expression 0 = 21x^20 + 11x^10 + 13 true. So there are no critical numbers, and therefore no local max or min.

Would this be correct?

 

[bblenyesi]

Yes, since for stationary/critical/etc... points to exist, your function's derivative has to have points in which its value is 0. Since your function can never have 0 values, you're correct.
The graphical interpretation is also quite neat. Try these in Mathematica, it'll all be clear in a second, and you can also use it in the case of more complicated functions:

$$Plot[x^{21} + x^{11} + {13*x}, \{ x, -10, 10\\\}]$$

$$Plot[21*x^{20} + 11*x^{10} + 13*x, \{ x, -10, 10\\\}]$$