relative extrema inflection point analysis

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10fehyrf

Mathematics

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1.

Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 6x(1/3) + 3x(4/3). You must justify your answer using an analysis of f '(x) and f "(x). (10 points)

2.

What is the maximum volume in cubic inches of an open box to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? Your work must include a statement of the function and its derivative. Give one decimal place in your final answer. (10 points)

3.

The position function of a particle in rectilinear motion is given by s(t) = t3 - 12t2 + 45t + 4 for t ≥ 0 with t measured in seconds and s(t) measured in feet. Find the position and acceleration of the particle at the instant when the particle reverses direction. Include units in your answer. (10 points)

4.

A circle is growing so that the radius is increasing at the rate of 3 cm/min. How fast is the area of the circle changing at the instant the radius is 12 cm? Include units in your answer. (10 points)

5.

The side of a square is measured to be 12 ft with a possible error of ±0.1 ft. Use linear approximation or differentials to estimate the error in the calculated area. Include units in your answer. (10 points)

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Explanation & Answer

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Hey there! Here are the answers 😇

Q1)
To find the relative extrema we first find the first derivative of the function and then
find the values of x that makes the first derivative zero or undefined. These values f x
are the critical values of the function.
d
(6𝑥1/3 + 3𝑥 4/3 ) = 2𝑥 −2/3 + 4𝑥1/3 = 0
dx
∴ 2𝑥 −2/3 (1 + 2𝑥) =

2(1 + 2𝑥)
=0
𝑥 2/3

∴ x=−

1
2

To make the derivative undefined the denominator equals to zero and hence, x = 0.
∴ x=−
We need to test the first derivative
-

2(1+2𝑥)
𝑥 2/3

1
𝑜𝑟 0
2

at values near this domain:

1

At negat...


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