Question

the smallest positive integer p for which expression x2-2px+3p+4 is negative for at least one real x​

Answer 1

given p is the smallest positive integer

so p=1

2x-2x+3+4 it gives positive

so p=2

2x-4x+6+4

=-2x+10 if we take x<=5

it gives positive and zero

if we take x>5

it gives negative

Answer 2

Answer:

The smallest positive integer p for which expression is negative is 5

Step-by-step explanation:

Let the given equation be

$f(x) = {x}^{2} - 2px + 3p + 4$

Given that the value of equation us negative.As the coefficient of first term is positive,the discriminant is also positive

$∆ > 0 \\ = > ( - 2p) {}^{2} - 4(1)(3p + 4) > 0 \\ = > 4 {p}^{2} - 12p - 16 > 0 \\ = > {p}^{2} - 3p - 4 > 0$

After factorisation,we get

$(p + 1)(p - 4) > 0$

From the above inequality we have

$p∈(−∞,−1)∪(4,∞)$

From this interval,the smallest positive integer is 5.

Therefore,The smallest positive integer p for which expression is negative is 5

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