# How do I find extrema of f(x) = x^7 - 14x^5 - 4x^3 - x^2 + 3 on a graphing calculator?

Jan 3, 2015

Fastest way would be to use the derivative function.

Whenever the derivative, also called the slope function reaches $0$ you have reached a top or 'valley' of your original function.

You can find the derivative by remembering:

if $f \left(x\right) = a \cdot {x}^{n}$ then $f ' \left(x\right) = n \cdot a \cdot {x}^{n - 1}$

While numbers without $x$ have a derivative of $0$

Going from there, the derivative of your function is:

$f ' \left(x\right) = 7 \cdot {x}^{7 - 1} - 5 \cdot 14 {x}^{5 - 1} - 3 \cdot 4 {x}^{3 - 1} - 2 \cdot {x}^{2 - 1} + 0$, or

$f ' \left(x\right) = 7 {x}^{6} - 70 {x}^{4} - 12 {x}^{2} - 2 x$

Enter this in your GC as $Y 1 =$
Now, on your GC you go find the zeroe-values (there's a menu for that)
Since your equation is 7th grade (because of ${x}^{7}$) you may find as many as 6 zero points for the derivative ($x = 0$ is one of them).
Fill in these $x$'s in the original equation and you will have the co-ordinates of all extremes.