[SOLVED] horizontal tangent line

**vinson24** · October 15th 2008, 08:08 AM

find the values of x at at which the curve f(x)=(3x-4)^7(x+5)^4 has a horizontal tangent line

**vinson24** · October 15th 2008, 08:59 AM

i know how to get two of the answers which are 4/3,-5 but the third answer -89/33 i dont know how to get

**Chris L T521** · October 15th 2008, 09:08 AM

Originally Posted by vinson24

find the values of x at at which the curve f(x)=(3x-4)^7(x+5)^4 has a horizontal tangent line

Find $f'(x)$

$f'(x)=21\left(3x-4\right)^6\left(x+5\right)^4+4\left(3x-4\right)^7\left(x+5\right)^3$

There are horizontal tangents when

$21\left(3x-4\right)^6\left(x+5\right)^4+4\left(3x-4\right)^7\left(x+5\right)^3=0$

This implies that $\left(3x-4\right)^6\left(x+5\right)^3\left[21\left(x+5\right)+4\left(3x-4\right)\right]=0$

So we see that either

$\left(3x-4\right)^6=0$ <----- here, you get $x=\tfrac{4}{3}$

$\left(x+5\right)^3=0$ <----- here, you get $x=-5$

$\left[21\left(x+5\right)+4\left(3x-4\right)\right]=0$ <----- solve this guy and see what you get

Can you take it from here?

--Chris

**Soroban** · October 15th 2008, 09:19 AM

Hello, vinson24!

I assume you know about horizontal tangents
and that it's the algebra that troubles you . . .

Find the values of $x$ at at which the curve $f(x)\:=\:(3x-4)^7(x+5)^4$
has a horizontal tangent line.

You know what to do, right? . . . Solve $f'(x) = 0$

Product Rule: . $f'(x) \;=\;(3x-4)^7\cdot4(x+5)^3 + (x+5)^4\cdot 7(3x-4)^6\cdot3 \;=\;0$

We have: . $4(3x-4)^7(x+5)^3 + 21(3x-4)^6(x+5)^4\;=\;0$

Factor: . $(3x-4)^6(x+5)^3\bigg[4(3x-4) + 21(x+5)\bigg] \;=\;0$

. . . . . . $(3x-4)^6(x+5)^3\bigg[12x - 16 + 21x + 105\bigg] \;=\;0$

. . . . . . . . . . . . . . $(3x-4)^6(x+5)^3(33x+89) \;=\;0$

And we have: . $\begin{Bmatrix}3x-4 &=& 0 & \quad\Rightarrow\quad & x &=& \boxed{\tfrac{4}{3}} \\ \\[-4mm]<br /> x + 5 &=& 0 & \Rightarrow & x &=& \boxed{\text{-}5} \\ \\[-4mm]<br /> 33x + 89 &=& 0 & \Rightarrow & x &=& \boxed{\text{-}\tfrac{89}{33}} \end{Bmatrix}$

But please check my algebra . . .
.

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