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Find all real values of m for which the inequalty mx 2 - 4x + 3m + 1 >0 is satisfied for all positive x.

Find all real values of m for which the inequalty mx2- 4x + 3m + 1 >0 is satisfied for all positive x.

Grade:11

1 Answers

Nishant Vora IIT Patna
askIITians Faculty 2467 Points
6 years ago
Hello Student, Please find the solution

The value of this quadratic is always positive (>0)
So graph will always be above x-axis
therefore the condition will be :-
D>0 and a>0 (parabola upwards and not touching the x-axis)

Solve both the conditions nw
1) D>0
16 – 4m(3m+1) >0
16 – 12m2 – 4m >0
multiplying by -1 both side
12m2+ 4m -16<0
Divide by 4
3m2+m – 4 <0
3m2+4m – 3m – 4 <0
m(3m+4) – 1 (3m + 4) <0
(3m+4)(m – 1) <0
m \epsilon [-4/3,1]

2) a>0
=>m>0

Now you can intersection of both the cases
m \epsilon (0,1]

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