Line $11x+3y-48 = 0$ tangent a graph $f(x) = \frac{4x + 3}{3x - 6}$ at $(a,b)$

Question

Line $11x+3y-48 = 0$ tangent a graph $f(x) = \frac{4x + 3}{3x - 6}$ at (a,b) when $a<b$

$a-b = ...$

Find gradient of the line, df(x) / dx

$\frac{(4x+3)3 - (3x-6)(4)} {(3x-6)^2}$

Which the same as $ -11/3$

$(12x+9 - 12x + 24 ) 3= -11(3x-6)^2$

If i solve x, literally i solve $a $ right? Is there less complicated way to solve it?

Looks your gradient has a flip in sign in the numerator. When using quotient rule it helps to remember the numerator always starts with the denominator function. — AgentS, Nov 23 '19 at 16:59

Dr. Sonnhard Graubner · Answer 1 · 2019-11-23T17:08:04.900

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The equation $$-\frac{11}{3}x+16=\frac{4x+3}{3x-6}$$ must have only on e solution. Compute the discriminant! It is $x=3$

edited Nov 23 '19 at 17:08

answered Nov 23 '19 at 17:01

Dr. Sonnhard Graubner

95,283

Sorry. The answer for (a,b) is e you meant? – Dini Nov 23 '19 at 17:09
Yes, it is $a=3$,and $$1\cdot 3+3\cdot b=48$$ – Dr. Sonnhard Graubner Nov 23 '19 at 17:11
But theres non in the options – Dini Nov 23 '19 at 19:32

score 0 · Accepted Answer · answered Nov 24 '19 at 13:21

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Rewrite $$f(x)=\frac43+\frac{11}{3x-6}.$$ Now $$f'(x)=-\frac{11}{(3x-6)^2}\cdot3$$ and $f'(x)=-11/3$ gives $(3x-6)=\pm1$, that is $x=7/3$ and $x=5/3$.

answered Nov 24 '19 at 13:21

Michael Hoppe

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How do come up with idea to make it $$f(x)=\frac43+\frac{11}{3x-6}.$$ – Dini Jan 30 '20 at 11:41
Because it makes the derivation easier. – Michael Hoppe Jan 30 '20 at 12:55
Yes i know the purpose. How do i can make a equation to be easier to differentiate – Dini Jan 30 '20 at 12:57
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I presume that there's no general rule. – Michael Hoppe Jan 30 '20 at 12:58

Line $11x+3y-48 = 0$ tangent a graph $f(x) = \frac{4x + 3}{3x - 6}$ at $(a,b)$

2 Answers2