# How do you find the extrema for f(x)=x^4-18x^2+7?

Jun 17, 2015

#### Answer:

This function has 3 extrema:

• A maximum at 0 $f \left(0\right) = 7$
• 2 minima at -3 and 3 $f \left(- 3\right) = f \left(3\right) = - 74$

#### Explanation:

To calculete the extrema of a function you have to find points, where $f ' \left(x\right) = 0$ first.
In this case you get:
$4 {x}^{3} - 36 x = 0$
$4 x \left({x}^{2} - 9\right) = 0$
$x = 0 \vee x = - 3 \times x = 3$

Now you have to check how $f ' \left(x\right)$ looks like in the surrounding of the points calculated above.
To check the behaviour you can either draw a graph or calculate $f ' ' \left(x\right)$

1. If $f '$ changes sign from positive to negative or $f ' ' < 0$ then it is a maximum
2. If $f '$ changes sign from negative to positive or $f ' ' > 0$- it is a minimum
3. If $f '$ does not change sign then there is no extremum at this point.