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

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

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Answer 1 · 2017-01-01T16:00:17

See the sketch of the graph below

Explanation:

We need

$(x^n)'=nx^(n-1)$

$a^2-b^2=(a-b)(a+b)$

Let $f(x)=x^4+2x^2-3$

We can factorise

$f(x)=(x^2+3)(x^2-1)=(x^2+3)(x+1)(x-1)$

$f'(x)=4x^3+4x=4x(x^2+1)$

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

$4x(x^2+1)=0$ for $x=0$

We can calculate the second derivative

$f''(x)=12x^2+4$

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

We can do a sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-1$ $color(white)(aaaaaaaa)$ $0$ $color(white)(aaaaaaaa)$ $1$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $f'(x)$ $color(white)(aaaaa)$ $-$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $0$ $color(white)(aaaa)$ $+$ $color(white)(aaaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ #color(white)(aaaa)↘# $color(white)(aaaaa)$ $-3$ $color(white)(aaaaaaaa)$ $↗$

$lim_(x->+-oo)f(x)=+oo$

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

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

1 Answer

Explanation:

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Science

Math

Humanities

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

1 Answer

Explanation:

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