The minute hand of a clock has a length of 12 cm, and the hour hand has a length of 8 cm

Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = x3 − 6x2 − 15x + 2 (a) Find the interval on which

Burka [1]

Answer:

a) $(-\infty, -1) \cup (5, \infty)$

b) $(-1,5)$

Step-by-step explanation:

The first step to solve this question is finding the roots of the derivative of x.

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

$ax^{2} + bx + c, a\neq0$ .

This polynomial has roots $x_{1}, x_{2}$ such that $ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})$ , given by the following formulas:

$x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}$

$x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}$

$\bigtriangleup = b^{2} - 4ac$

In this question:

$f(x) = x^{3} - 6x^{2} - 15x + 2$

So

$f'(x) = 3x^{2} - 12x - 15$

Finding the roots:

$3x^{2} - 12x - 15 = 0$

Simplifying by -3

$x^{2} - 4x - 5 = 0$

So $a = 1, b = -4, c = -5$

Then

$\bigtriangleup = (-4)^{2} - 4*1*(-5) = 36$

$x_{1} = \frac{-(-4) + \sqrt{36}}{2} = 5$

$x_{2} = \frac{-(-4) - \sqrt{36}}{2} = -1$

So the function can be divided in three intervals.

They are:

Less than -1

Between -1 and 5

Higher than 5

In which it increases and which it decreases?

Less than -1

Lets find the derivative in a point in this interval, for example, -2

$f'(x) = 3x^{2} - 12x - 15$

$f'(-2) = 3*(-2)^{2} - 12*(-2) - 15 = 21$

Positive.

So in the interval of $(-\infty, -1)$ , the function increases.

Between -1 and 5

Will choose 0.

$f'(x) = 3x^{2} - 12x - 15$

$f'(0) = 3*(0)^{2} - 12*(0) - 15 = -15$

Negative.

So in the interval of $(-1,5)$ , the function decreases.

Higher than 5

Will choose 6.

$f'(x) = 3x^{2} - 12x - 15$

$f'(6) = 3*(6)^{2} - 12*(6) - 15 = 21$

Positive

So in the interval of $(5, \infty)$ , the function increases.

(a) Find the interval on which f is increasing.

Using interval notation

$(-\infty, -1) \cup (5, \infty)$

b) Find the interval on which f is decreasing.

$(-1,5)$

5 0

3 years ago

The price of an item you want to buy is given in dollars and cents. You pay for it in cash by giving the clerk d dollars and c c

MrRa [10]

Answer:

$f(x,y) = \left \{ {(d-x,c-y) \, \, {y \, \leq c} \atop (d-x-1, 100+c-y) \, \, {y \, > \, c}} \right.$

Where x is the amount of dollars the item costs and y is the amount of cents. The first entry of f(x,y) is the amount of dollars to be returned and the second entry is the amount of cents to be returned.

Step-by-step explanation:

let f(x,y) be the function that computes the change.

The parameter x is the amount of dollars it costs and y is the amount of cents. both x and y should be integers and both of them should be positive or 0. Also y must be lower than 100, and x cant be bigger than d. Also if x = d, then it must happen that y ≤ c. The return value f(x,y) is a pair (a,b) where a and b satisfies the same conditions than the ordered pair (x,y) (the change cant be higher than the amount paid). a represents the amount of dollars that has to be returned and b the amount of cents.

Now, lets go to the formula. The formula should depend on wether y is higher than d or not, if that is the case, then after substracting x from d, we need to substract one more and add 100 cents to c-y (because we cant be given negative cents)

$f(x,y) = \left \{ {(d-x,c-y) \, \, {y \, \leq c} \atop (d-x-1, 100+c-y) \, \, {y \, > \, c}} \right.$

8 0

3 years ago

Where are negative numbers placed on a number line

ahrayia [7]

Answer: Negative numbers are placed to the left of zero on the number line

6 0

3 years ago

Read 2 more answers

Casadee is saving $17.00 each week to buy a new sewing machine that cost one hundred seventy five dollars and fifty cents how ma

Katyanochek1 [597]

About 11 weeks at the least