Question

what is the minimum value of 7^(x^2-4x+7)

Answer 1

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

Answer 2

yes

Answer 3

@mathmate @UnkleRhaukus

Answer 4

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

Answer 5

what do you mean by the "d"?

Answer 6

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

Answer 7

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

Answer 8

http://www.mememaker.net/static/images/templates/14288.jpg

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

Howw??

Answer 15

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

Answer 16

Completing the squares will do the job.

Answer 17

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

Answer 18

(x-2)^2=y-3

Answer 19

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

Answer 20

http://i1.kym-cdn.com/entries/icons/original/000/009/993/tumblr_m0wb2xz9Yh1r08e3p.jpg

Answer 21

Good job livelaughlilz! Speed+accuracy!

Answer 22

I like his movies! :)

Answer 23

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

Answer 24

Yes^ @livelaughlilz

Answer 25

7^2 = ???

Answer 26

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

Answer 27

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

Answer 28

You were thinking about Jackie Chan. :D

Answer 29

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

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

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

Answer 35

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

Answer 36

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.

Answer 37

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

Answer 38

@livelaughlilz is a boss at this.

Answer 39

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

Answer 40

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

Answer 41

oh

Answer 42

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

Answer 43

See you all!