F(x) = 3x + 2 and g(x) = x + 5 g[f(x)] =
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Answer:
Step-by-step explanation:
<u>Optimizing With Derivatives </u>
The procedure to optimize a function (find its maximum or minimum) consists in :
- Produce a function which depends on only one variable
- Compute the first derivative and set it equal to 0
- Find the values for the variable, called critical points
- Compute the second derivative
- Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum
We know a cylinder has a volume of 4 . The volume of a cylinder is given by
Equating it to 4
Let's solve for h
A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is
Replacing the formula of h
Simplifying
We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero
Rearranging
Solving for r
Computing h
We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative
We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.
The minimum area is
Angles 1, 2, and 3 are all 90°.
10x = 90
.. x = 9
x +y = 90
.. y = 81
5z = 90
.. z = 18
The values of (x, y, z) are (9, 81, 18).
10x = 90
.. x = 9
x +y = 90
.. y = 81
5z = 90
.. z = 18
The values of (x, y, z) are (9, 81, 18).