## mathcalculus Group Title help help help. Determine the extrema of f(x)= (-4)* x/ x^2+7 below on the given interval (a) on [1,4] The minimum is ?? and the maximum is ?? (b) on [1,5] The minimum is ?? and the maximum is ?? one year ago one year ago

1. mathcalculus Group Title

do we use the quadratic formula for this one?

2. mattt9 Group Title

did you find the derivative?

3. mathcalculus Group Title

derivative is 1*x^2+7-2x*(-4)*x/ (x^2+7)^2

4. mathcalculus Group Title

from there.. x^2-6x+7/x^4+14x^2+49

5. mathcalculus Group Title

then i = to 0

6. mathcalculus Group Title

and i'm left with x^2-6x+7

7. mathcalculus Group Title

however, i can't factor this... so do i use quadratic formula?

8. mattt9 Group Title

yeah I did not check your math, however if it is correct then use the quadratic formula.

9. mathcalculus Group Title

can you try the problem also?

10. mathcalculus Group Title

i want to know if its correct.

11. zepdrix Group Title

"derivative is 1*x^2+7-2x*(-4)*x/ (x^2+7)^2" That first term shouldn't be a 1. The derivative of -4x is not 1. I think that should fix it up for you.

12. mathcalculus Group Title

yes i must of typed it wrong. but that's exactly what i got here.

13. zepdrix Group Title

$\large f'(x)=\frac{\color{royalblue}{(-4)}(x^2+7)-(-4x)\color{royalblue}{(2x)}}{(x^2+7)^2}$

14. mathcalculus Group Title

then after..

15. zepdrix Group Title

Set it equal to zero, then multiply both sides by the denominator. Don't do long division or anything silly like that.

16. mathcalculus Group Title

yeah i ended up with x^2-6x+7/x^4+14x^2+49

17. mathcalculus Group Title

??

18. zepdrix Group Title

Sorry website wasn't working...

19. zepdrix Group Title

$\large 0=\frac{-4x^2-28+8x^2}{(x^2+7)^2}$Multiplying both sides by the denominator gives us,$\large 0=-4x^2-28+8x^2$ I don't understand how you got the x term in the middle.

20. mathcalculus Group Title

how do i get the derivative. can you show me?

21. zepdrix Group Title

$\large f(x)=\frac{-4x}{x^2+7}$ Remember the quotient rule, it will tell us to setup the derivative like this,$\large f'(x)=\frac{\color{royalblue}{(-4x)'}(x^2+7)-(-4x)\color{royalblue}{(x^2+7)'}}{(x^2+7)^2}$The blue terms are the ones we need to differentiate.

22. mathcalculus Group Title

yeah

23. mathcalculus Group Title

according to the quotient rule.

24. zepdrix Group Title

Which gives us this,$\large f'(x)=\frac{\color{royalblue}{(-4)}(x^2+7)-(-4x)\color{royalblue}{(2x)}}{(x^2+7)^2}$ Which simplifies to this,$\large f'(x)=\frac{-4x^2-28+8x^2}{(x^2+7)^2}$Right?

25. mathcalculus Group Title

oh ok thank u!

26. mathcalculus Group Title

then it's 4x^2-28 right?

27. zepdrix Group Title

Yes. From there you can find a critical point.

28. mathcalculus Group Title

29. mathcalculus Group Title

?

30. mathcalculus Group Title

@zepdrix

31. mathcalculus Group Title

i appreciate your help a lot. i just need time to figure this out... :(

32. zepdrix Group Title

Yah that sounds right. So we have a couple steps now. We plug our critical points into the original function, and write down the $$\large f$$ values they produce. Then, plug the end points into the original function, and write down the $$\large f$$ values they produce. Then simply compare the $$\large f$$ values. The largest will be your maximum. The smallest, your minimum.

33. zepdrix Group Title

$\large x \in\left[1,4\right]$These are our end points, 1 and 4. The end points of our interval.

34. mathcalculus Group Title

is the first critical point: -2 radical 7/7 ?

35. mathcalculus Group Title

thats the positive criticla point that i used.

36. mathcalculus Group Title

@zepdrix

37. zepdrix Group Title

yah that sounds right.

38. mathcalculus Group Title

39. mathcalculus Group Title

hope it's right :(

40. mathcalculus Group Title

are you checking to see if i am doing the problem correct?

41. mathcalculus Group Title

42. zepdrix Group Title

no the negative root should produce 2sqrt7/7 i think.

43. mathcalculus Group Title

?

44. mathcalculus Group Title

it's not $\frac{ -2\sqrt{7} }{ 7 }$ ??