What is the extreme (x^2+4x+3)/(x^3-2x^2-5x+6) ?

The extreme (x^2+4x+3)/(x^3-2x^2-5x+6) is Minimum(-6.69935…,-0.06007…),Maximum(1.85470…,-3.67281…)

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extreme (x^2+4x+3)/(x^3-2x^2-5x+6)

extreme points

\frac{x ^{2} + 4 x + 3}{x ^{3} - 2 x ^{2} - 5 x + 6}

Minimum (- 6.69935 \dots, - 0.06007 \dots), Maximum (1.85470 \dots, - 3.67281 \dots)

Solution steps

Find the critical points:

x \approx - 6.69935 \dots, x \approx 1.85470 \dots

Domain of

\frac{x ^{2} + 4 x + 3}{x ^{3} - 2 x ^{2} - 5 x + 6} : x < - 2 or - 2 < x < 1 or 1 < x < 3 or x > 3

Find intervals:

Decreasing

: - \infty < x < - 6.69935 \dots,

Increasing

: - 6.69935 \dots < x < - 2,

Increasing

: - 2 < x < 1,

Increasing

: 1 < x < 1.85470 \dots,

Decreasing

: 1.85470 \dots < x < 3,

Decreasing

: 3 < x < \infty

Plug the extreme point

x = - 6.69935 \dots

into

\frac{x ^{2} + 4 x + 3}{x ^{3} - 2 x ^{2} - 5 x + 6} \Rightarrow y = - 0.06007 \dots

Minimum (- 6.69935 \dots, - 0.06007 \dots)

Plug the extreme point

x = 1.85470 \dots

into

\frac{x ^{2} + 4 x + 3}{x ^{3} - 2 x ^{2} - 5 x + 6} \Rightarrow y = - 3.67281 \dots

Maximum (1.85470 \dots, - 3.67281 \dots)

Minimum (- 6.69935 \dots, - 0.06007 \dots), Maximum (1.85470 \dots, - 3.67281 \dots)

f (x, y) = 4 x^{3} + y^{3} - 12 x - 3 y

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

f (x) = 2 x^{2} - 4 x

f (x) = 12 x y - x^{3} - y^{3}

f (x, y) = 338 y^{2} + x^{2} - x^{2} y

What is the extreme (x^2+4x+3)/(x^3-2x^2-5x+6) ?
The extreme (x^2+4x+3)/(x^3-2x^2-5x+6) is Minimum(-6.69935…,-0.06007…),Maximum(1.85470…,-3.67281…)