Question

SOLVE AND SHOW COMPLETE SOLUTION

1. if y= ax^3 + bx^2 and its point of inflection is at (2,8), what are the values of a and b?

2. graph the curve and find the point of inflection: y= (9x^2 - x^3 + 6)/ (6).

3. graph the curve and find the point of inflection: y= x^3 - 3x^2 + 6.

Answer 1

1.

$$y= ax^3 + bx^2$$

$$y'=3 ax^2 +2 bx$$

$$y''=6ax+2b$$

$$y''(2)=0=>6a(2)+2b=0=>b=-6a$$

$$y(2)=8=>8a+4b=8=>b=-2a+2$$

$$-6a=-2a+2=>a=-\dfrac{1}{2}$$

$$b=-6(-\dfrac{1}{2})=3$$

$$y=-\dfrac{1}{2}x^3+3x^2$$

$$a=-\dfrac{1}{2}, b=3$$

2.

$$y= \dfrac{9x^2 - x^3 + 6}{6}$$

$D(y): (-\infin, \infin)$

$E(y): (-\infin, \infin)$

$y\to\infin$ as $x\to -\infin$

$y\to-\infin$ as $x\to \infin$

$y(0)=1$

$$y'= \dfrac{18x - 3x^2 }{6}= 3x-\dfrac{x^2}{2}$$

Find the critical number(s)

$$y'=0=>3x-\dfrac{x^2}{2}=0$$

$$x_1=0, x_2=6$$

Critical numbers: $0, 6.$

$$y(0)=1$$

$$y(6)= \dfrac{9(6)^2 - (6)^3 + 6}{6}=19$$

If $x<0, y'<0, y$ decreases.

If $0<x<6, y'>0, y$ increases.

If $x>6, y'<0, y$ decreases.

The function $y$ has the local maximum with value of $19$ at $x=6.$

The function $y$ has the local minimum with value of $1$ at $x=0.$

$$y''=3-x$$

$$y''=0=>3-x=0=>x=3$$

$$y(3)= \dfrac{9(3)^2 - (3)^3 + 6}{6}=10$$

If $x<3, y''>0, y$ is concave up.

If $x>3, y''<0, y$ is concave down.

Point $(3, 10)$ is the inflection point.

Graph the function

3.

$$y= x^3 - 3x^2 + 6$$

$D(y): (-\infin, \infin)$

$E(y): (-\infin, \infin)$

$y\to-\infin$ ​as$x\to -\infin$

$y\to\infin$ ​as$x\to \infin$

$y(0)=6$

$$y'=3x^2-6x$$

Find the critical number(s)

$$y'=0=>3x^2-6x=0$$

$$x_1=0, x_2=2$$

Critical numbers: $0, 2.$

$$y(0)=6$$

$$y(2)=(2)^3 - 3(2)^2 + 6=2$$

If $x<0, y'>0, y$ increases.

If $0<x<2, y'<0, y$ decreases.

If $x>2, y'>0, y$ increases.

The function $y$ has the local maximum with value of $6$ at $x=0.$

The function $y$ has the local minimum with value of $2$ at $x=2.$

$$y''=6x-6$$

$$y''=0=>6x-6=0=>x=1$$

$$y(1)=(1)^3 - 3(1)^2 + 6=4$$

If $x<1, y''<0, y$ is concave down.

If $x>1, y''>0, y$ is concave up.

Point $(1, 4)$ is the inflection point.

Graph the function