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Let i=inta^b(x^4-2x^2)dx for (a,b) which...

Let `i=int_a^b(x^4-2x^2)dx` for `(a,b)` which given integration is minimum `(bgt0)` (a) `(sqrt2,-sqrt2)` (b) `(0,sqrt2)` (c) `(-sqrt2,sqrt2)` (d) `(sqrt2,0)`

A

`(-sqrt(2),0)`

B

`(0,sqrt(2))`

C

`(sqrt(2),-sqrt(2))`

D

`(-sqrt(2),sqrt(2))`

Text Solution

Verified by Experts

The correct Answer is:
D
We have , I `=int_(a)^(b)(x^(4)-2x^(2))dx`
Let `f(x)=x^(4)-2x^(2)=x^(2)(x^(2)-2)`
` =x^(2)(x-sqrt(2))(x+sqrt(2))`
Graph ` y=f(x)=x^(4)-2x^(2)` is

Note that the definite integral `int_(a)^(b)(x^(4)-2x^(2)) dx` represent the area bounded by ` y= f(x),x =` a , b and the X - axis.
But between `x=-sqrt(2)andx=sqrt(2),f(x)` lies below the X - axis and so value definite integral will be negative. Also , as long as f(x) lie below the X - axis , the value of definite integral will be minimum.
`:.(a.b)=(-sqrt(2),sqrt(2)` for minimum of I.
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