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1 year ago

What is the y-intercept of the function /(x) = -2/9x + 1/3

Mathematics

1 answer:

IgorC [24]1 year ago

8 0

Answer:

Step-by-step explanation:

f(x)= -2/9x + 1/3 = -2/9(0) + 1/3 = 0 + 1/3= 1/3

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Find the absolute maximum and absolute minimum of the function f(x,y)=2x3+y4f(x,y)=2x3+y4 on the region {(x,y)|x2+y2≤64}{(x,y)|x

ddd [48]

Check where the first-order partial derivatives vanish to find any critical points within the given region:

$f(x,y)=2x^3+y^4\implies\begin{cases}f_x=6x^2=0\\f_y=4y^3=0\end{cases}\implies(x,y)=(0,0)$

The Hessian for this function is

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}12x&0\\0&12y^2\end{bmatrix}$

with $\det\mathbf H(0,0)=0$ , so unfortunately the second partial derivative test fails. However, if we take $x=0$ we see that $f(x,y)>0$ for different values of $y$ ; if we take $y=0$ we see $f(x,y)$ takes on both positive and negative values. This indicates (0, 0) is neither the site of an extremum nor a saddle point.

Now check for points along the boundary. We can parameterize the boundary by

$(x,y)=(8\cos t,8\sin t)$

with $0\le t\le2\pi$ . This turns $f(x,y)$ into a univariate function $F(t)$ :

$F(t)=f(8\cos t,8\sin t)=2^{10}\cos^3t+2^{12}\sin^4t$

$\implies F'(t)=3\cdot2^{10}\cos^2t(-\sin t)+2^{14}\sin^3t\cos t=2^{10}\sin t\cos t(16\sin^2t-3\cos t)$

$F'(t)=0\implies\begin{matrix}\sin t=0\implies t=0,\,t=\pi\\\\\cos t=0\implies t=\dfrac\pi2,\,t=\dfrac{3\pi}2\\\\16\sin^2t-3\cos t=0\implies t=2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3},\,t=2\pi-2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3}\end{matrix}$

At these critical points, we get

$F(0)=1024$

$F\left(2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3}\right)\approx893$

$F\left(\dfrac\pi2\right)=4096$

$F(\pi)=-1024$

$F\left(\dfrac{3\pi}2\right)=4096$

$F\left(2\pi-2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3}\right)\approx893$

We only care about 3 of these results.

$t=\dfrac\pi2\implies(x,y)=(0,8)$

$t=\pi\implies(x,y)=(-8,0)$

$t=\dfrac{3\pi}2\implies(x,y)=(0,-8)$

So to recap, we found that $f(x,y)$ attains

a maximum value of 4096 at the points (0, 8) and (0, -8), and
a minimum value of -1024 at the point (-8, 0).

6 0

2 years ago

The expression 16x2 + 4x + 125 represents the height, in feet, of a rock, x seconds after being thrown from a cliff. What does t

xeze [42]

The motion of the rock thrown from the cliff which goes downwards due to

the gravitational force, is a free fall motion.

125 represents; <u>A. The initial height of the rock</u>.

Reasons:

The given function that represents the height of the of the rock in feet <em>x</em> seconds after being thrown is; f(x) = 16·x² + 4·x + 125

At the initial height before being thrown from the cliff, we have;

The time, x = 0

The initial height is therefore;

The initial height f(0) = 16 × (0)² + 4 × (0) + 125 = 125

125 represents <u>A. The initial height of the rock</u>

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4 0

2 years ago

PLEASE ANSWER ASAP!!!!!!

Romashka [77]

The range in the average rate of change in temperature of the substance is from a low temperature of 1 F to a high of -11 F.

<h3>What is a formula for Fahrenheit?</h3>

The conversion formula for a temperature that is expressed on the Celsius (°C) scale to its Fahrenheit (°F) ;

°F = (9/5 × °C) + 32.

Given function:

f(x)= -6 sin(7/3 x+ 1/6) -5

The function will be maximum at the 7/3 x +1/6= 3π/2

So, the maximum temperature will be

= -6 sin (3π/2) -5

= 6 -5

= 1 F

The function will be minimum at the 7/3 x +1/6= π/2

Therefore, the maximum temperature will be

= -6 sin (π/2) - 5

= -6 -5

= -11 F

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5 0

1 year ago

Find the unknown length in the right triangle 27cm and 13 cm

Phoenix [80]

use pythagoras:

${c}^{2} = \: {b}^{2} - {a}^{2}$

c^2 = (27)^2 - (13)^2

c^2 = 560

c = 23.6643....cm

3 0

2 years ago

Write -3/4x-6 in standard form using integers

taurus [48]

Answer:

$3x+4y=-24$

Therefore, option C is correct.

Step-by-step explanation:

We have been given the equation:

$y=\frac{-3x}{4}-6[/$

We will take LCM 4 on right hand side of the above equation:

$y=\frac{-3x-24}{4}$

Now, we will multiply the 4 in denominator on right hand side to the y in left hand side pof the equation we get:

$4y=-3x-24$

After rearranging the terms we get:

$3x+4y=-24$

Therefore, option C is correct.

8 0

2 years ago