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

- anonymous

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

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- saifoo.khan

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

- anonymous

yes

- saifoo.khan

@mathmate @UnkleRhaukus

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## More answers

- saifoo.khan

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

- anonymous

what do you mean by the "d"?

- saifoo.khan

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

- anonymous

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

- saifoo.khan

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

- anonymous

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

- mathmate

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

- mathmate

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

- saifoo.khan

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

- KingGeorge

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

- saifoo.khan

Howw??

- mathmate

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

- mathmate

Completing the squares will do the job.

- anonymous

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

- anonymous

(x-2)^2=y-3

- mathmate

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

- saifoo.khan

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

- mathmate

Good job livelaughlilz! Speed+accuracy!

- mathmate

I like his movies! :)

- anonymous

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

- saifoo.khan

Yes^ @livelaughlilz

- mathmate

7^2 = ???

- saifoo.khan

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

- mathmate

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

- saifoo.khan

You were thinking about Jackie Chan. :D

- KingGeorge

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

- mathmate

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

- saifoo.khan

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

- anonymous

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

- mathmate

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

- anonymous

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

- anonymous

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

- mathmate

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.

- anonymous

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

- saifoo.khan

@livelaughlilz is a boss at this.

- mathmate

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

- mathmate

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

- anonymous

oh

- anonymous

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

- mathmate

See you all!

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