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avatar+69 

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)

 May 26, 2021
 #1
avatar+2207 
+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

 

=^._.^=

 May 26, 2021
 #3
avatar+780 
+1

Nice! I was dumb and did not know how to remove the abs.

MathProblemSolver101  May 26, 2021
 #2
avatar+780 
+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 $84+1=\boxed{85}$

 May 26, 2021

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