How do you find the exact relative maximum and minimum of the polynomial function of $2 x^{3} - 23 x^{2} + 78 x - 72$ $2x^3 -23x^2+78x-72$ ?

Question

How do you find the exact relative maximum and minimum of the polynomial function of $2 x^{3} - 23 x^{2} + 78 x - 72$ $2x^3 -23x^2+78x-72$ ?

1 Answer

Impact of this question

2285 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-11T09:34:02

There is a minimum at $x = 5.135$ $x=5.135$
There is a maximum at $x = 2.531$ $x=2.531$

Explanation:

Given -

$y = 2 x^{3} - 23 x^{2} + 78 x - 72$ $y=2x^3-23x^2+78x-72$

$\frac{d y}{d x} = 6 x^{2} - 46 x + 78$ $dy/dx=6x^2-46x+78$

$\frac{d^{2} y}{{d x}^{2}} = 12 x - 46$ $(d^2y)/(dx^2)=12x-46$

$\frac{d y}{d x} = 0 \Rightarrow 6 x^{2} - 46 x + 78 = 0$ $dy/dx=0 => 6x^2-46x+78=0$

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 \times a}$ $x=(-b+-sqrt(b^2 -4ac))/(2xxa)$

$x = \frac{- (- 46) \pm \sqrt{(- 46^{2}) - (4 \times 6 \times 78)}}{2 \times 6}$ $x=(-(-46)+-sqrt((-46^2) -(4xx6xx78)))/(2xx6)$

$x = \frac{46 \pm \sqrt{2116 - 1872}}{2 \times 6}$ $x=(46+-sqrt(2116-1872))/(2xx6)$

$x = \frac{46 \pm \sqrt{244}}{12}$ $x=(46+-sqrt(244))/(12)$

$x = \frac{46 \pm 15.62}{12}$ $x=(46+-15.62)/(12)$

$x = \frac{46 + 15.62}{12} = \frac{61.62}{12} = 5.135$ $x=(46+15.62)/(12)=61.62/12=5.135$

$x = \frac{46 - 15.62}{12} = \frac{30.38}{12} = 2.531$ $x=(46-15.62)/(12)=30.38/12=2.531$

At $x = 5.135$ $x=5.135$

$\frac{d^{2} y}{{d x}^{2}} = 12 (5.135) - 46 = 61.62 - 46 = 15.62 > 0$ $(d^2y)/(dx^2)=12(5.135)-46=61.62-46=15.62 >0$

There is a minimum at $x = 5.135$ $x=5.135$

At $x = 2.531$ $x=2.531$

$\frac{d^{2} y}{{d x}^{2}} = 12 (2.531) - 46 = 30.37 - 46 = - 15.628 < 0$ $(d^2y)/(dx^2)=12(2.531)-46=30.37-46=-15.628<0$

There is a maximum at $x = 2.531$ $x=2.531$

enter image source here

Science

Math

Humanities

... and beyond

How do you find the exact relative maximum and minimum of the polynomial function of $2 x^{3} - 23 x^{2} + 78 x - 72$ $2x^3 -23x^2+78x-72$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you find the exact relative maximum and minimum of the polynomial function of 2x3−23x2+78x−722x^3 -23x^2+78x-72?

1 Answer

Explanation:

Related questions

Impact of this question

How do you find the exact relative maximum and minimum of the polynomial function of $2 x^{3} - 23 x^{2} + 78 x - 72$ $2x^3 -23x^2+78x-72$ ?