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Question

(a) Given the a system of simultaneous equation below,
x + 3y + 3z = 3
x + 4y + 3z = 8
x + 4y + 4z = 2
i. Express the system in matrix form
ii. Using any of the matrix methods find the solutions simultaneous equations

(b) Find the maximum and minimum value of the function
2x3 - 3x2 – 12x + 8

Answer 1

1  3  3  3

1  4  3  8

1  4  4  2

is the matrix that you want, in augmented form.  I advise using the reduced row echelon method, available on my TI-83/84 calculator--just input this 3 by 4 matrix, carefully, and apply the rref function to it.

 

Alternatively, you can do row operations on the augmented matrix, manually...

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problem (b)  f(x) = 2x³ - 3x² -12x + 8

f'(x) = 6x² - 6x - 12; set this derivative to zero to find critical points.

6(x² - x -2) = 0

(x-2)(x+1) = 0

Critical x values are 2 and -1. 

 

Second derivative is f"(x) = 12x - 6

Since f"(2) is positive, the concavity there is upward, so f(2) is a local minmum;

since f"(2) is negative, the concavity there is downward, so f(-1) is a local maximum.

 

You can find those local extrema (y) values by evaluating the original function.