Home
Class 12
MATHS
If a x^2+b/xgeqc for all positive x wher...

If `a x^2+b/xgeqc` for all positive `x` where `a >0` and `b >0,` show that `27 a b^2geq4c^3dot`

Text Solution

Verified by Experts

Given, ` ax ^(2) + (b)/(x) ge c, AA x gt 0 , a, b gt 0`
Let `" " f (x) =a x ^(2) + (b)/(x) -c `
` therefore " " f' (x) = 2 ax - (b)/(x^(2)) ( 2 ax ^(2) - b)/( x ^(2))`
` rArr f '' (x) = 2 a + (2 b ) /(x ^(3)) gt 0 `[ since, a , b are all positive ]
Now, put ` f ' (x) = 0 rArr x = ((b )/(2a))^(1//3) gt 0 " " [ because a, b gt 0 ] `
At ` " " x = ((b )/( 2a)) ^(1//3), f '' (x) = + ve `
` rArr f (x) ` has minimum at ` x = ((b)/( 2a)) ^(1//3)`
and ` f (((b )/( 2a))^(1//3)) = a ((b)/( 2a)) ^(2//3) + (b)/((b // 2a) ^(1//3)) - c ge 0`
` " " = (( 2a)/(b)) ^(1//3) * ( 3b)/(2) - c ge 0`
` rArr " " (( 2 a )/(b)) ^(1//3) * ( 3b)/(2) ge c`
On cubing both sides, we get
` " " ( 2a)/(b) * ( 27 b ^(3))/( 8) ge c ^(3)`
`rArr " " 27 ab ^(2) ge 4 c ^(3)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If a ,b ,a n dc are positive and a+b+c=6, show that (a+1//b)^2+(b+1//c)^2+(c+1//a)^2geq75//4.

The equation a x^2+b x+c=0 has real and positive roots. Prove that the roots of the equation a^2x^2+a(3b-2c)x+(2b-c)(b-c)+a c=0 re real and positive.

If the length of a focal chord of the parabola y^2=4a x at a distance b from the vertex is c , then prove that b^2c=4a^3dot

If a , b , c are positive numbers such that a gt b gt c and the equation (a+b-2c)x^(2)+(b+c-2a)x+(c+a-2b)=0 has a root in the interval (-1,0) , then

The equation 4ax^2 + 3bx + 2c = 0 where a, b, c are real and a+b+c = 0 has

let f(x)=-x^3+(b^3-b^2+b-1)/(b^2+3b+2) if x is 0 to 1 and f(x)=2x-3 if x if 1 to 3 .All possible real values of b such that f (x) has the smallest value at x=1 ,are

If tanthetaa n dsectheta are the roots of a x^2+b x+c=0, then prove that a^4=b^2(b^2-4ac)dot

If the equation |x^2+b x+c|=k has four real roots, then A. b^2-4cgt0 and 0ltklt(4c-b^2)/ 4 B. b^2-4clt0 and 0ltklt(4c-b^2)/4 C. b^2-4cgt0 and kgt(4c-b^2)/4 D. none of these

If y = A cos 4 x + B sin 4 x, A and B are constants then Show that y_2 + 16y =0

If alpha is a real root of the quadratic equation a x^2+b x+c=0a n dbeta ils a real root of -a x^2+b x+c=0, then show that there is a root gamma of equation (a//2)x^2+b x+c=0 whilch lies between alpha & beta