The even solution of equation dy/dx=12 is the first term of an arithmetical sequence with common difference 4. Find a7 and a11. y=2x^3-3x^2

To determine the 7th and the 11th terms of the
arithmetical sequence, we'll have to determine the 1st
term.

From enunciation, we know that the first term of the
arithmetical sequence is also the solution of the equatin dy/dx =
12.

We'll differentiate the given y with respect to
x.

dy/dx =
d(2x^3-3x^2)/dx

dy/dx = 6x^2 -
6x

We'll re-write the
equation:

6x^2 - 6x = 12

We'll
divide by 6:

x^2 - x = 2

We'll
subtract 2:

x^2 - x - 2 =
0

We'll apply quadratic
formula:

x1 = [1 + sqrt(1 +
8)]/2

x1 = (1+3)/2

x1 =
2

x2 = (1-3)/2

x2 =
-1

The even solution is x =
2.

The first term of the arithmetical sequence is a1 =
2.

We'll write the general term of an arithmetical
sequence:

an = a1 + (n-1)*d, where d is the common
difference

a7 = a1 + 6d

a7 = 2
+ 6*4

a7 = 2 + 24

a7 =
26

a11  =a1 + 10d

a11 = 2 +
10*4

a11 = 2 + 40

a11 =
42

The 7th and the 11th terms of the
arithmetical sequence are:

a7
= 26 ; a11 = 42.