The value of 1/x - 4/x + 4 is least when x =

Question

The value of  \frac{1}{x} - \frac{4}{√x} + 4  is least when x =

A. 1/√6
B. 1/4
C. 1/√2
D. 1/2
E. 1

chetan2u · Accepted Answer

Since we have √x, X has to be positive, so let x=y^2..
So (1/x)-(4/√x)+4=(1/y^2)-(4/y)+4= \frac{4y^2-4y+1}{y^2}=\frac{(2y-1)^2}{y^2} ..
Least value here will be 0 when numerator is 0. (2y-1)^2 will be 0 at y=1/2..
Thus x=y^2=(1/2)^2=1/4..

B

If you are clueless, just see here X is, it is in the denominator...
To make the term lowest, we have to maximize the - term, that is -(4/√x).
And X is the denominator, so we have to make the denominator the least.
In choices 1/4 is the least.
Need not be true always, but will increase the probability of answering correctly if you have nowhere to go.

chrtpmdr · Answer

Can someone explain this in a bit more detail? The approach seems really advanced and is not something I would arrive logically. What am I missing

I had a approach where I just plugged in numbers and it was clear that it has to be either 1 / sqrt (6) or 1 / 4 From here it got a bit messy. But I estimated sqrt 6 as 2.50 and with a bit of guesstimating 1/4 gave the least value. Think I could not do it under time pressure with this calculation.

Kritisood · Answer

try to get the expression in the form of (a+b)^2 or (a-b)^2
we can see from the equation that 4/√x is nothing but 2*1/√x*2 (a=1/√x b=2)
hence we can rewrite the equation as  (\frac{1}{√x} -2)^2 
a perfect sq is always = or > than 0. since we need to find the lease value we will take it as = 0

(1/√x -2)^2 = 0 
1/√x = 2
x=1/4

JaySivakumar · Answer

for these kind of questions I try going backwards with the answer choices. is that alright?

Regor60 · Answer

Set 1/X = Y^2,

so 1/(X^.5) = Y

So the expression becomes:

Y^2-4Y+4. This factors into:

(Y-2)^2.

Since this is always nonnegative, its minimum value is 0 when:

Y=2

Since 1/X=Y^2, 1/X=4 and:

X=1/4

Posted from my mobile device

kayarat600 · Answer

Substituting answers seemed like the logical approach but it gave me 0 for 1/4 what am I doing wrong.

Krunaal · Answer

Nothing wrong, it should be 0 when you substitute x = 1/4; 0 is the least value of  \frac{1}{x} - \frac{4}{\sqrt{x}} + 4

kayarat600 · Answer

Dammit I missread the question as highest!

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The value of 1/x - 4/x + 4 is least when x =

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