We are differentiating $\displaystyle y$ with respect to $\displaystyle x$. So we can treat $\displaystyle z$ as a constant.

Therefore, the derivative becomes $\displaystyle y'=2cos(2x)+2xz$.

By plugging in $\displaystyle x=$ $\displaystyle \frac{\pi }{2}$ and $\displaystyle z=3$, we get $\displaystyle -2+3\pi$.

The max's and min's a function on a closed interval can occur either at local extrema, or the endpoints. Local extrema occur when the derivative is 0. First, taking the derivative,

$\displaystyle y' = 3x^2 - 2x = x(3x -2)$

We see that the extrema will occur at $\displaystyle x = 0$ and$\displaystyle x =\frac{2}{3}$. Here, we can either check to see which of these extrema are max's using the first or second derivative test, or we can just plug them into our function -- if one of them was actually a min, it just won't be our answer. 

If you choose to go the first route, you have $\displaystyle y'' = 3x-2 + 3x = 6x-2$

$\displaystyle y''(0) = -2$

$\displaystyle y''(\frac{2}{3}) = 2$

So we know that the extrema at 0 is a local max, and at 2/3 is a local min.

Testing our max and two endpoints, we have

$\displaystyle f(0) = 3$

$\displaystyle f(2) = 7$

So our maximum value is 7.

Let the dimensions of the picture be x and y.

$\displaystyle Area=xy$

Our constraint is

$\displaystyle 80=2x+4y$

Solve for x

$\displaystyle x=2y-40$

Let us substitute that value of x back into our area equation

$\displaystyle Area=(2y-40)\cdot y=2y^2-40y$

In order to maximize this equation, we need to find its derivative and set it equal to zero

$\displaystyle Area'=4y-40$

$\displaystyle 0=4y-40$

$\displaystyle 0=y-10$

$\displaystyle 10=y$

Now that we have found our first dimension, we need to find the second

$\displaystyle 80=2x+4y$

$\displaystyle 80=2x+4(10)$

$\displaystyle 80=2x+40$

$\displaystyle 40=2x$

$\displaystyle 20=x$

Then our dimensions are $\displaystyle 10$ inches and $\displaystyle 20$ inches.

Points of inflection are found when the second derivative of a function is equal to zero. For our function

$\displaystyle f(x)=xe^x-x-1$

The first derivative is:

$\displaystyle f'(x)=e^x+xe^x-1$

By the product rule.

Our second derivative then is:

$\displaystyle f'(x)=2e^x+xe^x$

Setting this equal to zero we obtain:

$\displaystyle 2e^x+xe^x=0$

$\displaystyle e^x(2+x)=0$

Since we know $\displaystyle e^x$ can never be equal to zero, we are only concerned with $\displaystyle 2+x=0$, so $\displaystyle x=-2$

Maximum's and minimums of functions can take place where the derivative is 0 or undefined. To find out where this occurs, we take the derivative.

$\displaystyle f'(x) = 4x^3 - 12x^2 -16x = 4x(x^2 - 3x -4) = 4x(x-4)(x+1)$

As you can see, the derivative will be 0 at $\displaystyle x = 0, -1, 4$.

At this point, we have three options:

We can plug them all into the original equation and see which are the biggest and which are the smallest.

We can use the first derivative interval test to see which are mins and which are maxes.

We can use the second derivative test to see which are mins and which are maxes.

As we're likely going to have to plug them all into the equation anyway, and the problem doesn't ask to classify the points as local mins or maxes, the first approach is the most time effective. Plugging in our points we find that $\displaystyle f(0) = 5$ ; $\displaystyle f(-1) = 2$ ;$\displaystyle f(4) = -123$

Before we determine the answer, it's important to remember that the global maxes and mins on a closed domain can occur at the endpoints, and on an open domain, may not exist. For this problem, we're given a quartic that faces upwards. Thus, there is no global maximum because the graph shoots off to infinity on either side. On the other hand, there is a global min because the graph is continuous and does not go down to negative infinity.

As we know the min must occur at one of the points we've determined above, we can see that we have a global min of $\displaystyle -123$ at $\displaystyle x=4$ and no global maximum.

Also note, as a bit of terminology, that a global maximum or minimum value is a value that occurs at a point. Thus our answer isn't that the maximum value is (4, -123), but just $\displaystyle -123$.

This question is an optimization problem. This question asks to find the distance to run along the beach that minimizes the time it takes to get to the swimmer in the water. To minimize time, we need to construct an equation, where time is a function of one variable. Fortunately, the equation

$\displaystyle Distance = Rate \cdot Time$

can be solved for time, and this is how we will create the equation we need.

Solving this equation for time gives

$\displaystyle Time = \frac{Distance}{Rate}$

We have two Rates. Running on the beach, and Swimming in the water. So we have two times to consider:

$\displaystyle Time_{Run} = \frac{Distance_{Run}}{Rate_{Run}}$ and $\displaystyle Time_{Swim} = \frac{Distance_{Swim}}{Rate_{Swim}}$

Adding these two times gives the total time to get to the person in the water.

$\displaystyle Time = \frac{Distance_{Run}}{Rate_{Run}} + \frac{Distance_{Swim}}{Rate_{Swim}}$

We are given the rates that the lifeguard can run and swim, 10ft/sec, and 4ft/sec respectively. The variable is how far to swim down the beach.

Labeling the distance the lifeguard runs as $\displaystyle x$ will make the math somewhat difficult, whereas labeling the distance the lifeguard runs as $\displaystyle 200-x$ will make the math a little nicer. We just need to remember that we did this. The picture below shows the labeling used in this explanation.

With this labeling, the distance the life guard runs is $\displaystyle 200-x$, and the distance the life guard swims is $\displaystyle \sqrt{x^2 + 100^2}$ by Pythagorean Theorem.

Plugging the distances and rates into our Time equation gives:

$\displaystyle Time = \frac{200-x}{10} + \frac{\sqrt{x^2 + 100^2}}{4}$

This expresses Time as a function of one variable, $\displaystyle x$. This is what we need to minimize. To do this we will find the relative minimum of this function. So we find the first derivative. First, we will do a little algebra and split the $\displaystyle \frac{200-x}{10}$ into two fractions and rewrite the square root as an exponent to make the derivative easier to compute.

$\displaystyle Time = \frac{200}{10} - \frac{x}{10} + \frac{\sqrt{x^2 + 100^2}}{4}$

$\displaystyle Time = 20 - \frac{1}{10}x + \frac{1}{4}(x^2 + 10000)^{1/2}$

Now we find the derivative. The derivative of $\displaystyle 20$ is $\displaystyle 0$.

The derivative of $\displaystyle -\frac{1}{10}x$ is $\displaystyle -\frac{1}{10}$.

We use the chain rule for $\displaystyle \frac{1}{4}(x^2 + 10000)^{1/2}$. Doing so gives

$\displaystyle \frac{1}{4}(\frac{1}{2})(x^2 + 10000)^{-1/2}(2x)$

Assembling the pieces results in the following derivative

$\displaystyle Time' = 0 - \frac{1}{10} + \frac{x}{4\sqrt{x^2 + 10000}}$

To find the critical points, we substitute 0 in for Time'.

$\displaystyle 0=- \frac{1}{10} + \frac{x}{4\sqrt{x^2 + 10000}}$

To solve for $\displaystyle x$, we move the  $\displaystyle -\frac{1}{10}$ to the opposite side and then cross multiply.

$\displaystyle \frac{1}{10}=\frac{x}{4\sqrt{x^2+10000}}$

$\displaystyle 4\sqrt{x^2 + 10000}= 10x$

 Now we can divide by 4 on both sides to isolate the square root, then reduce the resulting 10/4 to 5/2. Then we square both sides to eliminate the square root.

$\displaystyle \sqrt{x^2 + 10000}= \frac{5}{2}x$

$\displaystyle x^2 + 10000 = \frac{25}{4}x^2$

We know need to move the $\displaystyle x^2$ terms to the same side, the right side in this explanation Remember that we will need to get the common denominator to combine them.

$\displaystyle 10000 = \frac{25}{4}x^2 - x^2$

$\displaystyle 10000 = \frac{25}{4}x^2 - \frac{4}{4}x^2$

$\displaystyle 10000=\frac{21}{4}x^2$

Multiply both sides by $\displaystyle \frac{4}{21}$ to isolate $\displaystyle x^2$. Then square root both sides. We will not need to incorporate the $\displaystyle \pm$ when we square root, since $\displaystyle x$ is a physical distance.

$\displaystyle \frac{10000}{1} \cdot \frac{4}{21}= x^2$

$\displaystyle \frac{40000}{21}= x^2$

$\displaystyle \sqrt{\frac{40000}{21}}=x$

Since the question asks us to approximate the answer to the nearest hundredth, we can plug $\displaystyle \sqrt{\frac{40000}{21}}$ into a calculator to get a decimal. Doing so gives approximately x=43.64 ft.

However, this is not our final answer. Recall that we defined the distance the life guard ran as $\displaystyle 200-x$. So we need to subtract x=43.64 from 200 to find what the question is asking for. 

Doing this we get our final answer to be $\displaystyle 156.36 \text{ ft}$.

Since the rectangle has one corner on the origin, (0,0), and the opposite corner at a point (x,y) on the graph, we can set the width = x, and the length = y. 

The area of a rectangle is $\displaystyle A = (length)(width)$, so for this problem, $\displaystyle A=x\cdot y$.

Since we are trying to maximize the area, we need to express the area as a function of only one variable. Right now we have two variables. The way to fix this is to substitute one variable, like $\displaystyle y$, with an equivalent function of $\displaystyle x$.

Most problems give a relationship between the two variables. In this problem, the equation of the quarter-ellipse, $\displaystyle y = \sqrt{1 - \frac{1}{6}x^2}$, is the relationship we need.

We can substitute the $\displaystyle y$ in the Area equation with what is equals.

$\displaystyle A=x \cdot {\color{Red} \sqrt{1 - \frac{1}{6}x^2}}$

Now we have the equation of what we want to maximize, Area, written in terms of only one variable, x. Now we can find the relative maximum by finding the derivative.

Before finding the derivative, it will be helpful to rewrite the square root as an exponent.

$\displaystyle A= x(1 - \frac{1}{6}x^2)^{1/2}$

We will need to use the product rule, $\displaystyle {\color{Red} f'} \cdot {\color{Blue} g} + {\color{Red} g'} \cdot {\color{Blue} f}$, and use chain rule for $\displaystyle g'$. Doing this, we get:

$\displaystyle A' = {\color{Red} (1)}{\color{Blue} (1-\frac{1}{6}x^2)^{1/2} }+{\color{Red} (\frac{1}{2}(1 - \frac{1}{6}x^2)^{-1/2}(\frac{-2}{6}x))}{\color{Blue} (x)}$

Now we find the critical points of the derivative by setting it equal to zero and solving. We will simplify the derivative slightly at the same time. We can multiply the fractions in the second part, an move the $\displaystyle (1 - \frac{1}{6}x^2)^{-1/2}$ down to the denominator to remove the negative from the exponent. This gives us:

$\displaystyle 0 = (1 - \frac{1}{6}x^2)^{1/2} + \frac{-x^2}{6(1-\frac{1}{6}x^2)^{1/2}}$

Now we can solve for x. First, subtract the$\displaystyle (1 - \frac{1}{6}x^2)^{1/2}$ term to the left side.

$\displaystyle -(1 - \frac{1}{6}x^2)^{1/2} = \frac{-x^2}{6(1 - \frac{1}{6}x^2)^{1/2}}$

Now multiply both sides by denominator, $\displaystyle 6(1 - \frac{1}{6}x^2)^{1/2}$, in order to eliminate the fraction.

$\displaystyle -(1-\frac{1}{6}x^2)^{1/2}\cdot 6(1-\frac{1}{6}x^2)^{1/2}= -x^2$

The$\displaystyle (1 - \frac{1}{6}x^2)^{1/2}$ is multiplied by the same group, so we add their exponents, 1/2 + 1/2, which is 1. So we don't need to write the new exponent, since 1 is understood.

$\displaystyle -6(1-\frac{1}{6}x^2)= -x^2$

Since the power on the group is 1, we can distribute the -6 through it.

$\displaystyle -6 + x^2 = -x^2$

Moving all the $\displaystyle x^2$ terms to the left, and the -6 to the right, we get:

$\displaystyle 2x^2 = 6$

Divide both sides by 2 to isolate $\displaystyle x^2$

$\displaystyle x^2 = 3$

Now square root both sides.

$\displaystyle x = \pm \sqrt{3}$

Since the domain of the graph is given as $\displaystyle 0 \leq x \leq \sqrt{6}$, we can ignore the negative answer and use a positive answer.

Now we know the width is $\displaystyle x = \sqrt{3}$. To find the length, we need to find $\displaystyle y$. 

Plug $\displaystyle x = \sqrt{3}$ into the graph's equation to find $\displaystyle y$.

$\displaystyle y = \sqrt{1 - \frac{1}{6}x^2}$

$\displaystyle y = \sqrt{1 - \frac{1}{6}(\sqrt{3})^2}$

$\displaystyle y = \sqrt{1 - \frac{1}{6}(3)}$

$\displaystyle y = \sqrt{1 - \frac{1}{2}}$

$\displaystyle y = \sqrt{\frac{1}{2}}$

$\displaystyle y = \frac{1}{\sqrt{2}}$

We need to get the square root out of the denominator. To do this we multiply the numerator and denominator by $\displaystyle \sqrt{2}$.

$\displaystyle y = \frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}$

$\displaystyle y = \frac{\sqrt{2}}{2}$

Thus, the length of the rectangle should be $\displaystyle \frac{\sqrt{2}}{2}$.

Now we know the dimensions of the rectangle, that maximize its area:

width= $\displaystyle \sqrt{3}$

length =$\displaystyle \frac{\sqrt{2}}{2}$