How do you use the first and second derivatives to sketch y=2x^3- 3x^2 -180xy=2x33x2180x?

1 Answer
Dec 30, 2017

dy/dx(2x^3-3x^2-180x) = 6[x^2 - x -30]dydx(2x33x2180x)=6[x2x30]

(d^2y)/dx^2(2x^3-3x^2-180x) = 12x-6d2ydx2(2x33x2180x)=12x6

Graph is also available with relevant details.

Explanation:

We are given the function

color(red)(y = 2x^3-3x^2-180x)y=2x33x2180x

color(green)(Step.1)Step.1

We will find the First Derivative and set it equal to ZERO.

rArr dy/dx(2x^3-3x^2-180x)dydx(2x33x2180x)

rArr 2*dy/dx(x^3)-3*dy/dx(x^2)-180*dy/dx(x)2dydx(x3)3dydx(x2)180dydx(x)

rArr 2*3x^2-3*2x^1-180*123x232x11801

rArr 6x^2-6x - 1806x26x180

We will factor out the GCF

rArr 6*[ x^2-x - 30 ]6[x2x30]

This is the First Derivative of color(red)(y = 2x^3-3x^2-180x)y=2x33x2180x

Hence,

color(blue)((dy)/dx (2x^3-3x^2-180x) = 6*[ x^2-x - 30 ]dydx(2x33x2180x)=6[x2x30]

We will now set this equal to Zero.

:. 6*[ x^2-x - 30 ]= 0

rArr [ x^2-x - 30 ] = 0

We will split the x-term to get

rArr [ x^2+5x-6x - 30 ] = 0

rArr x*(x+5) - 6 *(x+5) = 0

rArr (x+5) *(x-6) = 0

Roots or Zeros are found at

color(blue)(x = -5 or x = +6)

We can now say that ...

the Critical Numbers in this graph are ( - 5 and 6 )

color(green)(Step.2)

Let us now place these critical values on a Number Line and then generate a Sign Chart

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Sign Chart for our First Derivative is given below:

enter image source here

We observe the following for the First Derivative Test:

  1. If the first derivative is Positive [ f'(x) > 0 ] ** then our Original Function is Increasing**.

  2. If the first derivative is Negative [ f'(x) < 0 ] ** then our Original Function is Decreasing**.

At . . color(red)(x = +6) our function has a maximum

At . . color(red)(x = -5) our function has a minimum

color(green)(Step.3)

We will now find the Second Derivative

We have

color(blue)((dy)/dx (2x^3-3x^2-180x) = 6*[ x^2-x - 30 ]

We must now find

color(blue)((d^2y)/dx^2 (2x^3-3x^2-180x)

color(blue)((dy)/dx [6*( x^2-x - 30 )]

That is we must differentiate our first derivative

rArr 6*[d/dx (x^2) +d/dx (-x) + d/dx (-30) ]

rArr 6*[(2x) -1 + 0]

rArr 6*[2x -1]

This is our Second Derivative

We will set the Second Derivative Equal to Zero

rArr 6*[2x -1] = 0

rArr [2x -1]= 0

rArr 2x = 1

rArr x = 1/2

color(blue)((d^2y)/dx^2 (2x^3-3x^2-180x) = 6*[2x -1]

Critical Point for the second derivative is 1/2

color(green)(Step.4)

Let us now place these values on a Number Line and then generate a Sign Chart

enter image source here

Sign Chart for our Second Derivative is given below:

enter image source here

We observe the following for the Second Derivative Test:

  1. If the second derivative is Positive [ f'(x) > 0 ] ** then our Original Function is Concave Up**.

  2. If the first derivative is Negative [ f'(x) < 0 ] ** then our Original Function is Concave Down**.

color(green)(Step.5)

The graph below is for the Original Function

color(red)(y = 2x^3-3x^2-180x)

enter image source here

Study the graph and compare the results obtained from the first derivative and the second derivative tests.

Hope this helps.