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 2 years ago
PLEASE HELP! :( The curve C has equation y={a(xa)^2}/x^24a^
where a is a positive constant. Show that C has one maximum point and one minimum point and find their coordinates.
 2 years ago
PLEASE HELP! :( The curve C has equation y={a(xa)^2}/x^24a^ where a is a positive constant. Show that C has one maximum point and one minimum point and find their coordinates.

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myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Find y' Set y' equal to 0 Solve y'=0 for x to get the critical numbers. Then we will determine if the critical numbers are max or mins or neither.

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0\[y={a(xa)^2}\div x^24a^2\]

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0I couldnot solve y'=0. I TRIED! :(

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2What did you get for y'

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Oh and we also need to find where y' does not exist

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0a(xa)^2(2x8a)(x^24a^2)(2ax2a^2)=0

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2hmmm.... You know a is a constant right?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Derivative of a is 0. Derivative of any constant *a is 0.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2a^2 is also a constant since a is a constant (a^2)'=0

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0Oops. Now I get it. O_o

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2And don't forget the quotient rules is : \[y'=\frac{ (\text{ derivative of top }) \cdot (\text{ bottom } )  ( \text{ derivative of bottom } ) \cdot ( \text{ top } )}{(bottom)^2}\]

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0Can you please show me what exactly y' would be? All these 'a' and brackets have confused me. :(

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0@experimentX : Can you please help me with this question?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2\[y=\frac{a(xa)^2}{x^24a^2}\] \[ \text{ What is } [a(xa)^2]' \text{ equal to? }\] \[ \text{ What is } [x^24a^2]' \text{ equal to?}\]

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2For my first question, a is a constant multiple. We can just bring that down and look at differentiating (xa)^2 by using chain rule.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2For my second question, 4a^2 is just a constant so we only need to look at differentiating x^2 by use of the power rule.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2When we get done, we will plug in the bottom, the top, the derivative of the bottom, and the derivative of the top into the following formula I gave you earlier: \[y'=\frac{ (\text{ derivative of top }) \cdot (\text{ bottom } )  ( \text{ derivative of bottom } ) \cdot ( \text{ top } )}{(bottom)^2} \]

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2So so far we have: \[y'=\frac{ (\text{ derivative of top }) \cdot (x^24a^2)  ( \text{ derivative of bottom } ) \cdot ( a(xa)^2 )}{(x^24a^2)^2} \]

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2I'm just asking you now to give me the derivative of \[ (a(xa))^2 \text{ and } (x^24a^2) \].

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0For (a(xa))^2, it would be 2x2a. For (x^24a^2), it would be 2x.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2For the first one, couldn't you just leave it as 2(xa) but don't forget to bring down that a in front. :) So you would actually have what for the first one?

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0Would it be 2a(xa)? :O

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Yep so we have \[y'=\frac{ (\text{ derivative of top }) \cdot (x^24a^2)  ( \text{ derivative of bottom } ) \cdot ( a(xa)^2 )}{(x^24a^2)^2} \] = \[y'=\frac{ ( 2a(xa) ) \cdot (x^24a^2)  ( 2x ) \cdot ( a(xa)^2 )}{(x^24a^2)^2}\] = \[=\frac{2a(xa)(x^24a^2)2xa(xa)^2}{(x^24a^2)^2}\]

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Now what factors in the numerator do the two terms on top have in common?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Yes. :) So factor that out.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2\[2a(xa)(x^24a^2)2xa(xa)^2\] = \[2a(xa)[ ?  ?]\]

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.02a(xa){(x^24a^2)x(xa)}

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Yep You will have \[\frac{2a(xa)[(x^24a^2)x(xa)]}{(x^24a^2)^2}\] You can simplify what you have in brackets above though.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2\[\frac{2a(xa)(x^24a^2x^2+xa)}{(x^24a^2)^2}\] You have like terms. :)

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Do you see the like terms to combine?

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0Yep. I'll end up with 2a(xa)(xa4a^2). Right?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Ok great now set equal factor equal to zero You are almost there :)

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2a is a positive constant so 2a can never be zero so set xa equal to zero and set xa4a^2=0 Solve both for x to find the critical numbers :)

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0So I'll have x=a and x=4a? What do I do now?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2It said find the coordinates so find what y is if x=a and find what y is if x=4a

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0If x=a, then y=0. If x=4a, y=9a^3. Is it correct?

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0No. Sorry. For x=4a, y is undefined.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.2Your first one is right. Your second one is wrong.

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0So y is not undefined?

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0Oops. I found out my mistake. For x=4a, y=3a/4. Thank you soooooo muchhhhh @myininaya ! :D

emcrazy14
 2 years ago
Best ResponseYou've already chosen the best response.0One more thing, how do I show which is the minimum point and which is maximum? Is it by the second derivative?
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