Minimum of Quadratic
Find the minimum value of 4 x 2 + 8 x + 1 6 for real x .
The answer is 12.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
4 solutions
Challenge Taking student note: Simple standard solution.
Method 2
Derrivative: 8x +8 =0 => x = -1
(# 2nd Derrivative 8>0 => minimum(unnnessary) )
Your explanation is wrong. The quadratic expression does not have a minimum just because x = − 1 . Or were you trying to say that it is minimum at x = − 1 ?
Log in to reply
its Very Clear that i suggested the value of x(= -1) or min(f(x))=12
Log in to reply
I was not referring to your answer. I was referring to the original solution
@Hung Woei Neoh The answer provided is absolutely correct. The quadratic expression attains minimum value at x= -1 because ( x + 1 ) 2 has a minimum value of 0, which occurs at x = -1.
He wasn't referring to the minimum value of the expression, but value of x that minimizes the function.
Log in to reply
The way he phrased it was wrong. It should be " minimum at x = − 1 ", not "minimum as x = − 1 ". These two phrases have completely different meanings.
Okay, my bad for nitpicking about language and word usage here
Log in to reply
@Hung Woei Neoh – Oh yep, yep, My bad. It does have a different meaning. Thanks, I edited the solution :)
Min value of a quadratic function
= 4 a − D
= 1 2
Here's the derivation.
Log in to reply
Let the min value be k.
a x 2 + b x + c = k
a x 2 + b x + c − k = 0
There's only 1 value of x for the min value k. So, the discriminant of the above equation should be 0.
b 2 − 4 a ( c − k ) = 0
k = c − 4 a b 2
k = 4 a 4 a c − b 2
Differentiate equation You get 8x+8 Equate to zero as minimum point Answer is -1 Substitute in original equation Answer is 12
Just draw the graph and find the coordinates of the vertex. The ordinate of the vertex is the required answer (=12).
Take 4 as common 4 ( x 2 + 2 x + 4 )
Re write as 4 [ ( x + 1 ) 2 + 3 ]
Now clearly it is minimum at x = − 1
So minimum value is 4 × 3 = 1 2