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What is the equation of the tangent to $y=x^3-6x^2+12x+2$ that is parallel to the line $y=3x$ ?

I have no idea, how to solve, no example is given in the book! Appreciate your help!

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    Calculus sure does help a lot here. The calculus-free approach is a little bit involved...2011-08-25

3 Answers 3

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Hint: what is the slope of $y=3x$? Then what is the slope of $y=x^3-6x^2+12x+2$ (which depends upon $x$)? You need to find an $x$ where the slopes match. Then find a $y$ so that $(x,y)$ is a point on the cubic. Now you have a point and a slope, giving you the equation for the line.

Added: the slope of the tangent is y'=3x^2-12x+12\ \ , which we are told is $3\ \ $. Solving $3=3x^2-12x+12\ \ $ gives $x=1 \text{ or } 3\ \ $ . So the points of tangency are $(1,9)$ and $(3,11)\ \ $. The lines with slope $3$ that pass through these points are $y=3x+6\ $ and $y=3x+2\ \ $. A figure is at Wolfram Alpha

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    great! I like it! Th$x$ alot Ross. I'm very happy! God $b$less you all! ;)2011-08-26
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The equation of the tangent to the graph of $f(x)$ at $(a,f(a))$ is given by

$y=f(a)+f^{\prime }(a)(x-a)=f^{\prime }(a)x+f(a)-f^{\prime }(a)a.\tag{1}$

Two lines with equations $y=mx+b$ and $y=m^{\prime }x+b^{\prime }$ are parallel if and only if $m=m^{\prime }$. Hence, the family of lines parallel to the line $y=3x$ is given by $y=3x+b$. So, we must have

$f^{\prime }(a)x+f(a)-f^{\prime }(a)a=3x+b.\tag{2}$

Equating coefficients we get $f^{\prime }(a)=3$ and $f(a)-f^{\prime }(a)a=b$. Since the derivative of $f(x)=x^{3}-6x^{2}+12x+2$ at $x=a$ is $f^{\prime }(a)=3a^{2}-12a+12$, we obtain the system of two equations

$3a^{2}-12a+12=3,\tag{3}$

$a^{3}-6a^{2}+12a+2-3a=b,$

which is equivalent to

$a=1,b=6\tag{4}$

or

$a=3,b=2.\tag{5}$

Hence the equations of the two tangent lines are

$y=3x+6\tag{6}$

and

$y=3x+2.\tag{7}$

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    @Sb Sangpi: Glad I could help.2011-08-26
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Look at a taylor series (ref 1, 2) of your function about a point $x_c$ of order 1 (linear).

Then find which $x_c$ produces a line parallel to $3x$, or has a slope of $3$. Hint! There are two solutions actually.

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    Yes - but I'm not saying *which* curve.2011-08-27