Question

1. find the maximum or minimum of the function: $f(x)=x^{2}-2x - 7$
2. you have 400 feet of fencing to enclose a rectangular region. what is the maximum area of the region? (justify your answer)

Explanation:

Step1: Find the derivative of $f(x)=x^{2}-2x - 7$

Using the power - rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=2x-2$.

Step2: Set the derivative equal to zero to find critical points

$2x - 2=0$, solving for $x$ gives $x = 1$.

Step3: Find the second - derivative of $f(x)$

$f''(x)=(2x - 2)'=2$. Since $f''(1)=2>0$, the function has a minimum at $x = 1$.

Step4: Calculate the minimum value of $f(x)$

Substitute $x = 1$ into $f(x)$: $f(1)=1^{2}-2\times1 - 7=1 - 2 - 7=-8$.

For the fencing problem:
Let the length of the rectangle be $x$ and the width be $y$. The perimeter of the rectangle (amount of fencing) is $P = 2x+2y$. Given $P = 400$, so $2x + 2y=400$, which simplifies to $y = 200 - x$.
The area of the rectangle $A(x)=x\times y=x(200 - x)=200x-x^{2}$.

Step1: Find the derivative of $A(x)$

$A'(x)=(200x - x^{2})'=200 - 2x$.

Step2: Set the derivative equal to zero to find critical points

$200 - 2x=0$, solving for $x$ gives $x = 100$.

Step3: Find the second - derivative of $A(x)$

$A''(x)=(200 - 2x)'=-2$. Since $A''(100)=-2<0$, the function has a maximum at $x = 100$.

Step4: Calculate the maximum area

Substitute $x = 100$ into $A(x)$: $A(100)=100\times(200 - 100)=10000$ square feet.

Answer:

1. The minimum value of $f(x)=x^{2}-2x - 7$ is $-8$.
2. The maximum area of the rectangular region with 400 feet of fencing is 10000 square feet.