Question

49 multiple choice 2 points
what is the value of x in the solution of the system of equations ( 3x + 2y = 12 ) and ( 5x - 2y = 4 )?
8
4
2
3
50 multiple choice 2 points
let f be defined by ( f(x)=(x^{2}-1)^{4} ) for all real numbers x. for what values of x is the function increasing?
( (-1,1) )
( (-1,0) ) and ( (1,infty) )
( (-infty,-1) ) and ( (1,infty) )
( (1,infty) )

Explanation:

49. Solving the system of equations \(3x + 2y=12\) and \(5x - 2y = 4\)

Step1: Add the two equations

Adding \(3x + 2y=12\) and \(5x - 2y = 4\) to eliminate \(y\).
\((3x + 2y)+(5x - 2y)=12 + 4\)
\(3x+5x+2y - 2y=16\)
\(8x=16\)

Step2: Solve for \(x\)

Divide both sides of \(8x = 16\) by \(8\).
\(x=\frac{16}{8}=2\)

50. Finding where \(f(x)=(x^{2}-1)^{4}\) is increasing

Step1: Find the derivative using the chain - rule

Let \(u=x^{2}-1\), then \(y = u^{4}\).
By the chain - rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
\(\frac{dy}{du}=4u^{3}\) and \(\frac{du}{dx}=2x\), so \(f^{\prime}(x)=4(x^{2}-1)^{3}\cdot2x=8x(x^{2}-1)^{3}=8x(x - 1)^{3}(x + 1)^{3}\)

Step2: Find the critical points

Set \(f^{\prime}(x)=0\). Then \(8x(x - 1)^{3}(x + 1)^{3}=0\), so \(x=-1,0,1\)

Step3: Use the test - intervals

• For \(x\in(-\infty,-1)\), let \(x=-2\). Then \(f^{\prime}(-2)=8\times(-2)\times((-2)-1)^{3}\times((-2)+1)^{3}=8\times(-2)\times(-27)\times(-1)<0\)
• For \(x\in(-1,0)\), let \(x =-\frac{1}{2}\). Then \(f^{\prime}(-\frac{1}{2})=8\times(-\frac{1}{2})\times((-\frac{1}{2})-1)^{3}\times((-\frac{1}{2})+1)^{3}=8\times(-\frac{1}{2})\times(-\frac{27}{8})\times(\frac{1}{8})>0\)
• For \(x\in(0,1)\), let \(x=\frac{1}{2}\). Then \(f^{\prime}(\frac{1}{2})=8\times\frac{1}{2}\times((\frac{1}{2})-1)^{3}\times((\frac{1}{2})+1)^{3}=8\times\frac{1}{2}\times(-\frac{1}{8})\times(\frac{27}{8})<0\)
• For \(x\in(1,\infty)\), let \(x = 2\). Then \(f^{\prime}(2)=8\times2\times(2 - 1)^{3}\times(2 + 1)^{3}=8\times2\times1\times27>0\)

The function \(f(x)\) is increasing when \(f^{\prime}(x)>0\), which is on the intervals \((-1,0)\) and \((1,\infty)\)

Answer:

1. C. 2
2. B. \((-1,0)\) and \((1,\infty)\)