Question

If the inequality (x-(a-1))(x-(a^2+2))<0 holds for all x£ (-1,3] then correct statement(s) is/are 1)a^2>=1 2)a>=1 3)a<=-1 4)a<=0 see I know that the answer is (1) and (3) but I need proper explanation for it​

Answer 1

SOLUTION

TO DETERMINE

If the inequality $\sf{(x - (a - 1))(x - ( {a}^{2} + 2)) < 0}$ holds for all x ∈ ( - 1 , 3 ] then correct statement(s) is/are

1) a² ≥ 1

2) a ≥ 1

3) a ≤ - 1

4) a ≤ 0

EVALUATION

Here the given inequality is

$\sf{(x - (a - 1))(x - ( {a}^{2} + 2)) < 0}$

Clearly a² + 2 > a - 1

So the above inequality holds when

$\sf{(a - 1) < x < ( {a}^{2} + 2)}$

Now it is given that - 1 < x ≤ 3

Thus we get

$\sf{a - 1 < - 1 \: \: \: and \: \: {a}^{2} + 2 \geqslant 3}$

$\sf{ \implies \: a < 0 \: \: \: and \: \: {a}^{2} \geqslant 1}$

$\sf{ \implies \: a < 0 \: \: \: and \: \: \bigg(a \geqslant 1 \: \: or \: \: a \leqslant - 1 \bigg)}$

Thus above holds when a ≤ - 1

FINAL ANSWER

Hence the correct option is 3) a ≤ - 1

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