What is the minimum value of p and q in the equation x3−px2+qx−8=0 where p and q are positive real numbers and roots of the equation are real?
What is the minimum value of p and q in the equation x3−px2+qx−8=0 where p and q are positive real numbers and roots of the equation are real?
The correct option is B 6,12
A cubic equation can be expressed in terms of its roots as
x3-(sum of the roots) x2+(sum of the product of the roots taken 2 at a time) x-(product of the roots)=0
thus,
p= sum of roots taken one at a time
q= sum of roots taken two at a time
Product of roots =8 (given in question)
'if the product of 2 or more numbers is constant, the sum will be minimum when they are equal'
i.e.If abc=constant, the minimum value of a+b+c will be obtained at a=b=c
Thus, For the sum of roots (p) to be minimum, 8 should be the product of 3 numbers that are equal to each other (difference between them should be equal). This number is 2. The roots are thus 2,2,2
P= 2+2+2= 6
And q= 2(2)+2(2)+2(2)= 12
Answer= option b
6,12
A cubic equation can be expressed in terms of its roots as
x3-(sum of the roots) x2+(sum of the product of the roots taken 2 at a time) x-(product of the roots)=0
thus,
p= sum of roots taken one at a time
q= sum of roots taken two at a time
Product of roots =8 (given in question)
'if the product of 2 or more numbers is constant, the sum will be minimum when they are equal'
i.e.If abc=constant, the minimum value of a+b+c will be obtained at a=b=c
Thus, For the sum of roots (p) to be minimum, 8 should be the product of 3 numbers that are equal to each other (difference between them should be equal). This number is 2. The roots are thus 2,2,2
P= 2+2+2= 6
And q= 2(2)+2(2)+2(2)= 12
Answer= option b
