Mathematics, 26.09.2019 08:50 bryan519688

The function g is given by g(x)=4x^3+3x^2-6x+1. what is the absolute minimum value of g on the closed interval [-2,1]?

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Answer from: crossoverking07

The smallest value of the output of the given function on the interval [-2, 1]

is the absolute minimum of the function in the interval.

The absolute minimum on the closed interval [-2, 1] is $\underline{(-2, -7)}$

Reasons:

The given function is g(x) = 4·x³ + 3·x² - 6·x + 1

At the extremum (maximum and minimum) points, we have;

$\dfrac{d\left(g(x) \right)}{dx} = \dfrac{d\left(4 \cdot x^3 + 3 \cdot x^2 - 6 \cdot x + 1\right)}{dx} = 12 \cdot x^2 + 6 \dot x - 6 = 0$

2·x² + x - 1 = 0

x² + 0.5·x - 0.5 = 0

Using a graphing calculator, we get;

(x + 1)·(x - 0.5) = 0

x = -1, and x = 0.5 both values are in the given interval [-2, 1]

At x = -1, g(-1) = 4×(-1)³ + 3×(-1)² - 6×(-1) + 1 = 6

At x = (0.5), g(0.5) = 4×(0.5)³ + 3×(0.5)² - 6×(0.5) + 1 = -0.75

Therefore;

The local minimum on the closed interval [-2, 1] is (0.5, -0.75)

However from plotting the graph, we have the following table of values;

$\begin{array}{|cc|c|}x&&g(x)\\-2&&-7\\-1&&6\\0&&1\\1&&2\end{array}\right]$

The absolute minimum is therefore, the point $\underline{(-2, -7)}$

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Answer from: Maaaryyss

First, find the derivative g '(x), set it = to zero, and solve for x (these values are called "critical values"). There will be 2 solutions / critical values.

Next, evaluate the given function at the endpoints of the given interval: at x=-2, x=1. Next, evaluate the given function at each of the 2 critical values you've just found (noting that they MUST be in the interval (-2,1) ). You will now have 4 y-values. Which one is the very smallest? That function value is your answer.
The function g is given by g(x)=4x^3+3x^2-6x+1. what is the absolute minimum value of g on the close

The function g is given by g(x)=4x^3+3x^2-6x+1. what is the absolute minimum value of g on the close

Answer from: Quest

2 & 7

step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

Answer from: Quest

step-by-step explanation:

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