What is the extreme f(x)=x^4+6x^3+24x^2+8x+1 ?

The extreme f(x)=x^4+6x^3+24x^2+8x+1 is Minimum(-0.17808…,0.30358…)

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extreme f(x)=x^4+6x^3+24x^2+8x+1

extreme points

f (x) = x^{4} + 6 x^{3} + 24 x^{2} + 8 x + 1

Minimum (- 0.17808 \dots, 0.30358 \dots)

Solution steps

Find the critical points:

x \approx - 0.17808 \dots

Domain of

x^{4} + 6 x^{3} + 24 x^{2} + 8 x + 1 : - \infty < x < \infty

Find intervals:

Decreasing

: - \infty < x < - 0.17808 \dots,

Increasing

: - 0.17808 \dots < x < \infty

Plug the extreme point

x = - 0.17808 \dots

into

x^{4} + 6 x^{3} + 24 x^{2} + 8 x + 1 \Rightarrow y = 0.30358 \dots

Minimum (- 0.17808 \dots, 0.30358 \dots)

f (x) = 3 x^{2} - 3 y^{2}

f (x, y) = x^{4} - 8 x^{3} + 22 x^{2} - 24 x

f (x, y) = - 4 x^{2} y + 4 x y^{2}

f (x, y) = x^{2} + y^{3} - 300 x - 75 y - 3

0.285 x^{2} - 2.28 x + 7.6

What is the extreme f(x)=x^4+6x^3+24x^2+8x+1 ?
The extreme f(x)=x^4+6x^3+24x^2+8x+1 is Minimum(-0.17808…,0.30358…)