# How do you sketch the graph y=x^4+2x^2-3 using the first and second derivatives?

Jan 1, 2017

See the sketch of the graph below

#### Explanation:

We need

$\left({x}^{n}\right) ' = n {x}^{n - 1}$

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Let $f \left(x\right) = {x}^{4} + 2 {x}^{2} - 3$

We can factorise

$f \left(x\right) = \left({x}^{2} + 3\right) \left({x}^{2} - 1\right) = \left({x}^{2} + 3\right) \left(x + 1\right) \left(x - 1\right)$

$f ' \left(x\right) = 4 {x}^{3} + 4 x = 4 x \left({x}^{2} + 1\right)$

We find critical points when $f ' \left(x\right) = 0$

$4 x \left({x}^{2} + 1\right) = 0$ for $x = 0$

We can calculate the second derivative

$f ' ' \left(x\right) = 12 {x}^{2} + 4$

$f ' ' \left(0\right) = 4 > 0$, this is a minimum

We can do a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$f ' \left(x\right)$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$color(white)(aaaa)↘$\textcolor{w h i t e}{a a a a a}$$- 3$$\textcolor{w h i t e}{a a a a a a a a}$↗

${\lim}_{x \to \pm \infty} f \left(x\right) = + \infty$

graph{(y-(x^4+2x^2-3))=0 [-7.9, 7.9, -3.95, 3.95]}