1. Solve for x and y:

|x|+x+y=14

x+|y|-y=12

2.All the roots of the cubic polynomial f(x)=3x^3+bx^2+c+d are negative real numbers. If f(0)=81, find the smallest possible value of f(1). Prove that your answer is the minimum

3.Let f(x) be an even function, and let g(x) be an odd function, such that f(x)+g(x)=x^2+2^x for all real numbers x. Find f(2)

imbadatmath May 26, 2021

#1**+2 **

|x|+x+y=14

x+|y|-y=12

Both x and y are positive.

2x + y = 14

x = 12

y = -10

WRONG (since -10 isn't positive)

x is positive and y is negative

2x + y = 14

x - 2y = 12

4x + 2y = 28

5x = 40

x = 8, y = -2

CORRECT

=^._.^=

catmg May 26, 2021

#2**+2 **

(2) $c+d=81$

$f(1) = 3 + b + c + d = 3 + b + 81 = 84 + b.$ $b$ is at least $1$ so the answer is $84+1=\boxed{85}$

MathProblemSolver101 May 26, 2021