Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I need help with the following question.

Find the largest positive value of x at which the curve:

$$y = (2x + 7)^6 (x - 2)^5$$

has a horizontal tangent line.

share|improve this question
    
Hint: Horizontal tangent line - is $\frac{dy}{dx} 0, $ or positive or negative? –  Siddhi V Iyer Mar 19 '12 at 20:49
    
A sketch often helps - easy to do here by first sketching the sextic and quintic factors. –  Mark Bennet Mar 19 '12 at 20:58
    
How do I proceed after finding the derivative of y? –  Xavier Mar 19 '12 at 21:37

3 Answers 3

To solve $y'(x)=0$ when $y(x)\ne0$, one can consider the derivative $\dfrac{y'(x)}{y(x)}$ of the function $\log|y(x)|=6\log|2x+7|+5\log|x-2|$. The computations become much simpler since $$ \frac{y'(x)}{y(x)}=\frac{6\cdot2}{2x+7}+\frac5{x-2}. $$ Thus, $y'(x)=0$ as soon as the RHS is zero, that is, when $12(x-2)+5(2x+7)=0$, that is, when $x=-\frac12$. Complete the reasoning with the values where $y(x)=0$, that is, $x=-\frac72$ and $x=2$.

share|improve this answer
    
Thanks for the insight, really makes the computation much simpler. –  Xavier Mar 19 '12 at 22:15
    
@Didier Can I find you in the chat? –  Pedro Tamaroff Mar 19 '12 at 22:17

Hint: what slope corresponds to a horizontal tangent? How do you find the slope of a tangent line?

share|improve this answer
    
Am I on the right track to first find the derivative of y? What do I do next? –  Xavier Mar 19 '12 at 20:56
    
@Xavier: Yes, you are on the right track. As Kirthi says, you want the derivative to be zero, as that is the slope of the tangent. –  Ross Millikan Mar 19 '12 at 21:44

Hint: $\displaystyle{\frac{dy}{dx} = \left(12(2x+7)^5(x-2)^5+5(2x+7)^6(x-2)^4 \right) = 0}$ , in other words

$\displaystyle{\frac{dy}{dx} = (2x+7)^5(x-2)^4(22x+11) = 0}$ at what points?

$\displaystyle{\frac{dy}{dx} = 0}$ at $x=-\frac{7}{2}, x=2, x=-\frac{1}{2}$

share|improve this answer
    
Hi, shouldn't the derivative be 12(2x+7)^5(x-2)^5 + 5(x-2)^4(2x+7)^6 ? Taking into account of chain-rule. Do correct me if I am wrong. –  Xavier Mar 19 '12 at 21:20
    
@Xavier: You are correct. –  Joe Johnson 126 Mar 19 '12 at 21:33
    
@JoeJohnson126: Thanks. Do you have any idea how to proceed from there? –  Xavier Mar 19 '12 at 21:36
    
@Xavier, corrected that but the approach is still the same. (I just gave a hint) –  Kirthi Raman Mar 19 '12 at 21:44

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.