find-minimun-and-maksimum-y-x-3-1-x-2-3-

Question Number 121368 by abdullahquwatan last updated on 07/Nov/20

find minimun and maksimum

y = \frac{x^{3} - 1}{x^{2} - 3}

Answered by MJS_new last updated on 07/Nov/20

x^{3} - 1 = 0 \Rightarrow zero at x = 1

x^{2} - 3 \neq 0 \Rightarrow asymptotes x = \pm \sqrt{3}

\Rightarrow - \infty < y < + \infty

\frac{x^{3} - 1}{x^{2} - 3} = x + \frac{3 x - 1}{x^{2} - 3} \Rightarrow asymptote y = x

y^{'} = \frac{x (x^{3} - 9 x + 2)}{{(x^{2} - 3)}^{2}}

y^{″} = \frac{6 (x^{3} - x^{2} + 9 x - 1)}{{(x^{2} - 3)}^{3}}

solving y^{'} = 0 and testing y^{″} {\begin{cases} < 0 (maximum) \\ = 0 (flat point) \\ > 0 (minimum) \end{cases}

leads to

x_{1} \approx - 3.10548 local \max with y \approx - 4.65822

x_{2} = 0 local minimum with y = \frac{1}{3}

x_{3} \approx . 223462 local maximum with y \approx . 335193

x_{4} \approx 2.88202 local minimum with y \approx 4.32303

Commented by abdullahquwatan last updated on 07/Nov/20

thank you sir

Commented by MJS_new last updated on 07/Nov/20

you are welcome

Answered by TANMAY PANACEA last updated on 07/Nov/20

\frac{d y}{d x} = \frac{(x^{2} - 3) (3 x^{2}) - (x^{3} - 1) 2 x}{{(x^{2} - 3)}^{2}}

\frac{d y}{d x} = \frac{3 x^{4} - 9 x^{2} - 2 x^{4} + 2 x}{{(x^{2} - 3)}^{2}} = \frac{x^{4} - 9 x^{2} + 2 x}{{(x^{2} - 3)}^{2}}

f o r m a x / m i n \frac{d y}{d x} = 0 x (x^{3} - 9 x + 2) = 0

x = 0 a n d 0.223, 2.882

w h e n x > 2.882 \frac{d y}{d x} > 0

w h e n 2.882 > x > 0.223 \frac{d y}{d x} < 0

w h e n 0.223 > x > 0 \frac{d y}{d x} > 0

w h e n x < 0 \frac{d y}{d x} < 0

s i g n o f \frac{d y}{d x} c h a n g e s f r o m - v e t o + v e w h e n c u r v e c r o s s x = 0

s o a t x = 0 g i v e n f u n c t i o n m i n i m u m

y_{x = 0} = \frac{1}{3} (m i n i m u m) = \frac{1}{3}

s i g n c h a n g e f r o m - v e t o + v e w h e n c r o s s x = 0.223

s o m a x i m u m y_{x = 0.223} = \frac{{(0.223)}^{3} - 1}{{(0.223)}^{2} - 3} (m a x i m u m) =

w h e n 2.882 > x \frac{d y}{d x} < 0

w h e n x > 2.882 \frac{d y}{d x} > 0

s i g n c h a n g e f r o m - v e t o + v e

s o m i n i m u m

y_{x = 2.882} = \frac{{(2.882)}^{3} - 1}{{(2.882)}^{2} - 3} (m i n i m u m) =

Commented by TANMAY PANACEA last updated on 07/Nov/20

p l s c a l c u l a t e t h e [v a l u e

Commented by abdullahquwatan last updated on 07/Nov/20

thank you sir

Commented by TANMAY PANACEA last updated on 07/Nov/20

m o s t w e l c o m e