For any quadratic function $y = - x^{2} + 2 x - 3, y > 0 \forall$

x \in ϕ

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x \in R

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x \in R - (0}

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Solution

The correct option is A $x \in ϕ$
Given the quadratic equation $y = - x^{2} + 2 x - 3$
Now, comparing the equation with $y = a x^{2} + b x + c,$ we get:
$a = - 1; b = 2; c = - 3$
We have $a = - 1 < 0$
This means the graph is a downward opening parabola.
Also, Discriminant $D = b^{2} - 4 a c$
$\Rightarrow D = 2^{2} - 4 \times - 1 \times - 3 \Rightarrow D = 4 - 12 = - 8 < 0$

Thus, the function has $a < 0$ as well as $D < 0.$
So, we can plot the function as:

Thus, $\forall x \in R, y < 0$
$Or y > 0 \forall x \in ϕ$

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