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

• BrownianMan
In summary, the function f(x) = x^21 + x^11 + 13x does not have a local maximum or minimum because its derivative can never equal 0 due to the even exponents of its terms. This can also be seen graphically by plotting the function and its derivative in Mathematica.
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?

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:

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

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

## What is a local maximum/minimum?

A local maximum/minimum is a point on a graph where the function reaches its highest/lowest value within a specific interval, but not necessarily the highest/lowest value overall.

## How do you determine if f(x) has a local maximum/minimum?

To determine if f(x) has a local maximum/minimum, you must first take the derivative of the function and set it equal to zero. Then, solve for x to find the critical points. Finally, use the second derivative test or a sign chart to determine if the critical points are local maxima or minima.

## What does it mean if f(x) has no local maximum/minimum?

If f(x) has no local maximum/minimum, it means that there are no points on the graph where the function reaches its highest/lowest value within a specific interval. This could indicate that the function is increasing or decreasing without any significant changes in direction.

## Can a function have more than one local maximum/minimum?

Yes, a function can have more than one local maximum/minimum. This can occur when the function has multiple peaks or valleys within a specific interval. In this case, each peak or valley would be considered a separate local maximum/minimum.

## How is the concept of local maximum/minimum important in mathematical analysis?

The concept of local maximum/minimum is important in mathematical analysis because it helps us understand the behavior of a function within a specific interval. It allows us to identify important points on a graph and determine the direction of the function at those points. This information is crucial in solving optimization problems and understanding the overall behavior of a function.

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