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3 years ago

If a and b are positive numbers, find the maximum value of f(x) = xa(7 â x)b on the interval 0 â¤ x â¤ 7.

1 answer:

8 0

Given $f(x)=x^a(7-x)^b$ on the interval 0 ≤ x ≤ 7, for maximum value, f'(x) = 0.

$f'(x)=0 \\ \\ \Rightarrow -bx^{a}(7-x)^{b-1}+ax^{a-1}(7-x)^b=0 \\ \\ \Rightarrow ax^{a-1}(7-x)^b=bx^{a}(7-x)^{b-1} \\ \\ \Rightarrow a(7-x)=bx \\ \\ \Rightarrow (a+b)x=7a \\ \\ \Rightarrow x= \frac{7a}{a+b}$

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Find the vertices and foci of the hyperbola. 9x2 − y2 − 36x − 4y + 23 = 0

Xelga [282]

Hey there, hope I can help!

NOTE: Look at the image/images for useful tips
$\left(h+c,\:k\right),\:\left(h-c,\:k\right)$

$\frac{\left(x-h\right)^2}{a^2}-\frac{\left(y-k\right)^2}{b^2}=1\:\mathrm{\:is\:the\:standard\:equation\:for\:a\:right-left\:facing:H}$
with the center of (h, k), semi-axis a and semi-conjugate - axis b.
NOTE: H = hyperbola

$9x^2-y^2-36x-4y+23=0 \ \textgreater \ \mathrm{Subtract\:}23\mathrm{\:from\:both\:sides}$
$9x^2-36x-4y-y^2=-23$

$\mathrm{Factor\:out\:coefficient\:of\:square\:terms}$
$9\left(x^2-4x\right)-\left(y^2+4y\right)=-23$

$\mathrm{Divide\:by\:coefficient\:of\:square\:terms:\:}9$
$\left(x^2-4x\right)-\frac{1}{9}\left(y^2+4y\right)=-\frac{23}{9}$

$\mathrm{Divide\:by\:coefficient\:of\:square\:terms:\:}1$
$\frac{1}{1}\left(x^2-4x\right)-\frac{1}{9}\left(y^2+4y\right)=-\frac{23}{9}$

$\mathrm{Convert}\:x\:\mathrm{to\:square\:form}$
$\frac{1}{1}\left(x^2-4x+4\right)-\frac{1}{9}\left(y^2+4y\right)=-\frac{23}{9}+\frac{1}{1}\left(4\right)$

$\mathrm{Convert\:to\:square\:form}$
$\frac{1}{1}\left(x-2\right)^2-\frac{1}{9}\left(y^2+4y\right)=-\frac{23}{9}+\frac{1}{1}\left(4\right)$

$\mathrm{Convert}\:y\:\mathrm{to\:square\:form}$
$\frac{1}{1}\left(x-2\right)^2-\frac{1}{9}\left(y^2+4y+4\right)=-\frac{23}{9}+\frac{1}{1}\left(4\right)-\frac{1}{9}\left(4\right)$

$\mathrm{Convert\:to\:square\:form}$
$\frac{1}{1}\left(x-2\right)^2-\frac{1}{9}\left(y+2\right)^2=-\frac{23}{9}+\frac{1}{1}\left(4\right)-\frac{1}{9}\left(4\right)$

$\mathrm{Refine\:}-\frac{23}{9}+\frac{1}{1}\left(4\right)-\frac{1}{9}\left(4\right) \ \textgreater \ \frac{1}{1}\left(x-2\right)^2-\frac{1}{9}\left(y+2\right)^2=1 \ \textgreater \ Refine$
$\frac{\left(x-2\right)^2}{1}-\frac{\left(y+2\right)^2}{9}=1$

Now rewrite in hyperbola standardform
$\frac{\left(x-2\right)^2}{1^2}-\frac{\left(y-\left(-2\right)\right)^2}{3^2}=1$

$\mathrm{Therefore\:Hyperbola\:properties\:are:}\left(h,\:k\right)=\left(2,\:-2\right),\:a=1,\:b=3$
$\left(2+c,\:-2\right),\:\left(2-c,\:-2\right)$

Now we must compute c
$\sqrt{1^2+3^2} \ \textgreater \ \mathrm{Apply\:rule}\:1^a=1 \ \textgreater \ 1^2 = 1 \ \textgreater \ \sqrt{1+3^2}$

$3^2 = 9 \ \textgreater \ \sqrt{1+9} \ \textgreater \ \sqrt{10}$

Therefore the hyperbola foci is at $\left(2+\sqrt{10},\:-2\right),\:\left(2-\sqrt{10},\:-2\right)$

For the vertices we have $\left(2+1,\:-2\right),\:\left(2-1,\:-2\right)$

Simply refine it
$\left(3,\:-2\right),\:\left(1,\:-2\right)$
Therefore the listed coordinates above are our vertices

Hope this helps!

8 0

4 years ago

If a triangle has a heig of 14 inches and base of 9 inches whats its area?

vlada-n [284]

Answer:

63 inches

Step-by-step explanation:

the formula for the area of a triangle is

a = 1/2 bh

plug in the numbers

a = 1/2(9)(14)

9 times 14 is 126

a = 1/2(126)

half of 126 (or 126 divided by 2)

area = 63 inches

8 0

3 years ago

Which animal's heart beats at a rate of 65 beats per minute?

loris [4]

Answer:

A cow

Step-by-step explanation:

7 0

3 years ago

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please help me with my sisters homework answers all of the questions please and show work I need it for today thanks

yawa3891 [41]

For 16 it's 8/10 for 17 it's 52/100 for 18 it's -92/100 for 19 it's -48/100 for 20 it's 86/100 for 21 it's 76/100

6 0

3 years ago

Ivahew [28]

Answer: 1/7

Step-by-step explanation:

Total questions answered = 49

Harry correct answers = 42

Harry wrong answers = (49 - 42) = 7

Thus, the only fraction of wrong answers = (Number of wrong answers / Total number of questions answered)

i.e 7/49

= 1/7

Thus, harry answered 1/7 of the 49 questions wrongly.

8 0

3 years ago

Read 2 more answers