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## livelaughlilz Group Title what is the minimum value of 7^(x^2-4x+7) one year ago one year ago

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1. saifoo.khan Group Title

$\Large 7^{x^2-4x+7}$Like this?

2. livelaughlilz Group Title

yes

3. saifoo.khan Group Title

@mathmate @UnkleRhaukus

4. saifoo.khan Group Title

Minimum value should be there where dy/dx = 0.

5. livelaughlilz Group Title

what do you mean by the "d"?

6. saifoo.khan Group Title

In other words minimum value occurs where the derivative of the function is equal to zero.

7. livelaughlilz Group Title

isn't derivatives calculus? i haven't learned that yet and i don't think we're expected to know it to solve this problem

8. saifoo.khan Group Title
9. livelaughlilz Group Title

haha i was thinking about it and wouldn't it be one over infinity? because if x^2-4x+7 was negative infinity than 7^(x^2-4x+7) would be 1/infinity

10. mathmate Group Title

Hint: 7^x is an increasing function. So if we minimize x, 7^x is also a minimum.

11. mathmate Group Title

x^2-4x+7 cannot be at -inf. It is a parabola concave upwards.!

12. saifoo.khan Group Title

And are we going to try that by trial and error? @mathmate

13. KingGeorge Group Title

No, we need to simply find the vertex of the parabola to find the minimum value of the parabola.

14. saifoo.khan Group Title

Howw??

15. mathmate Group Title

Good point!] @livelaughlilz can you take it from here?

16. mathmate Group Title

Completing the squares will do the job.

17. livelaughlilz Group Title

yeah the vertex of the parabola is (2,3)

18. livelaughlilz Group Title

(x-2)^2=y-3

19. mathmate Group Title

x^2-4x+7=(x-2)^2+3 so the vertex is at (2,3)

20. saifoo.khan Group Title
21. mathmate Group Title

Good job livelaughlilz! Speed+accuracy!

22. mathmate Group Title

I like his movies! :)

23. livelaughlilz Group Title

oh so the answer is 343. the lowest value of x is 2, and if you plug it in you get 7^3=343

24. saifoo.khan Group Title

Yes^ @livelaughlilz

25. mathmate Group Title

7^2 = ???

26. saifoo.khan Group Title

@mathmate : haha, nice. My brother just love his movies. i don't. :/

27. mathmate Group Title

Nevermind, I was out of my mind. Yes, 7^3=343

28. saifoo.khan Group Title

You were thinking about Jackie Chan. :D

29. KingGeorge Group Title

As a sidenote @mathmate, you can find the x-coordinate of the vertex using the formula $v_x=\frac{-b}{2a}$

30. mathmate Group Title

Thank you for the defence, even though it was a lame one! Good thinking! :)

31. saifoo.khan Group Title

I simply love this method. Satellite73 taught me this. -b/2a

32. livelaughlilz Group Title

to add onto @KingGeorge the y-coordinate of the vertex is -[(b^2-4ac)/(4a)]

33. mathmate Group Title

Thank you @KingGeorge, my memory is limited and sometimes defective. But thanks for the shortcut anyway!

34. livelaughlilz Group Title

you can figure all that out by completing the square of y=ax^2+bx+c

35. livelaughlilz Group Title

and making y=ax^2+bx+c into vertex form

36. mathmate Group Title

Actually I know the -b/2a part, that's how I get to do the completing the square. It's just I never managed to memorize the -[(b^2-4ac)/(4a)] part.

37. livelaughlilz Group Title

well b^2-4ac is also the discriminant...

38. saifoo.khan Group Title

@livelaughlilz is a boss at this.

39. mathmate Group Title

Perhaps from this time on I will remember. Practice makes perfect! :)

40. mathmate Group Title

Actually what I do is finding it by c-(b/2a)^2, which is exactly what we do when we complete the square.

41. livelaughlilz Group Title

oh

42. livelaughlilz Group Title

i'm going to close this question now. thanks everyone!

43. mathmate Group Title

See you all!