Minimum value?
Given that x > − 1 , find the minimum value of
6 ( x + 1 ) 4 x 2 + 8 x + 1 3
There's a easier way!
The answer is 2.
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.
3 solutions
why do you substitute a=x+1? Or how do you know to do this?
Log in to reply
You could substitute anything, but x-1 works here the best. It just takes some practice to see it.
Shouldn't it be minimized at p = 1?
@Trevor Arashiro , There's a typo in your solution. Minimization is attained at p = 1 , not maximization.
p is not maximized at 1 rather it should be minimized at 1.
6 ( x + 1 ) 4 x 2 + 8 x + 1 3 = x + 1 2 3 + 2 3 x + 1
And since x>-1, both are positive, you can use AM>=GM to find the minimum value
Did you use Heaviside?
differentiate and equate to zero and then find the value of x. we get two values for x, but one is less than -1. for more accurate result, check the condition for minima ie second derivative at this value of x should be greater than 0
Substitute a = x + 1
We are left with
6 a 4 a 2 + 9
6 a 4 a 2 + 6 a 9
3 2 a + 2 a 3
Substitute p = 3 2 a
p + p 1
This is obviously maximized at p=1 (or you could use am-gm).
Therefore, p = 3 2 a = 1
a = 2 3
x + 1 = 2 3
x = 2 1
Plugging x back in, we find the original function to equal 2.