The minimum value of fx-4x4-32x2-734x2-4 for -x-2- is

Given : f(x)=4x^4 32x^2+734(x^2 4) Dividing numerator by denominator, f(x) reduced to : f(x)=(x^2 4)+94(x^2 4) and (x^2 4) gt;0 for |x| gt;2 , using A.M. ≥ ...

You visited us 5 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Physics

Dimensional Analysis

The minimum v...

Question

The minimum value of $f (x) = \frac{4 x^{4} - 32 x^{2} + 73}{4 (x^{2} - 4)}$ for $| x | > 2,$ is

Open in App

Solution

Given : $f (x) = \frac{4 x^{4} - 32 x^{2} + 73}{4 (x^{2} - 4)}$
Dividing numerator by denominator, $f (x)$ reduced to :
$f (x) = (x^{2} - 4) + \frac{9}{4 (x^{2} - 4)}$
and $(x^{2} - 4) > 0$ for $| x | > 2,$
using $A . M . \geq G . M . :$
$\Rightarrow \frac{1}{2} [(x^{2} - 4) + \frac{9}{4 (x^{2} - 4)}] \geq \sqrt{(x^{2} - 4) \cdot \frac{9}{4 (x^{2} - 4)}} \Rightarrow (x^{2} - 4) + \frac{9}{4 (x^{2} - 4)} \geq 2 \times \frac{3}{2} \Rightarrow (x^{2} - 4) + \frac{9}{4 (x^{2} - 4)} \geq 3$
So, minimum value of $f (x)$ is $3.$

Suggest Corrections

Similar questions

Q. Let

f (x) = 4 x^{2} - 4 a x + a^{2} - 2 a + 2

be a quadratic polynomial in

x

a \in R

. If x-coordinate of vertex of parabola

y = f (x)

is less than

0

and

f (x)

has minimum value

3

for

x \in [0, 2]

, then the value of

a

Q. The minimum value of

f (x) = x + \frac{4}{x + 2}

Q. Let

f (x) = 4 x^{2} - 4 a x + a^{2} - 2 a + 2

be a quadratic polynomial in

x

a \in R

. If

y = f (x)

takes minimum value of

3

[0, 2]

and x-coordinate of vertex is greater than

2

, then value of

a

Q. Let F(x) = max {(x + 3), (7 – 2x)}. What is the minimum value of F(x) for 2

\geq

\geq

Q. For real

x

find the minimum value of

4 x^{2} - 2 x + 1

Join BYJU'S Learning Program

Explore more

Dimensional Analysis

Standard XII Physics

Join BYJU'S Learning Program

The minimum value of f(x)=4x4−32x2+734(x2−4) for |x|>2, is

The minimum value of $f (x) = \frac{4 x^{4} - 32 x^{2} + 73}{4 (x^{2} - 4)}$ for $| x | > 2,$ is