Let A be range of function $f (x) = (x^{2} + 2 x + 3)^{2} - 4 (x^{2} + 2 x + 3) + 6$ . Then which of the following sets the subsets of A?

[4, \infty]

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[1, 4]

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(4, 6)

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[6, 10]

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Solution

The correct option is A $[4, \infty]$
for $g (x) = x^{2} + 2 x + 3$
$g^{'} (x) = 2 x + 2 = 0 ⟹ x = - 1$
so, minima at $x = - 1$
and minimum value of $g (- 1) = 2$
so range of $g (x) = [2, \infty]$
Let $g (x) = z$
so $f (x) = z^{2} - 4 z + 6$
$f^{'} (x) = 2 z - 4 = 0 ⟹ z = 4$
and minimum value $f (2) = 4$
for range of $f (x) = A = [4, \infty]$

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Let A be range of function f(x)=(x2+2x+3)2−4(x2+2x+3)+6. Then which of the following sets the subsets of A?

The correct option is A [4,∞]for g(x)=x2+2x+3g′(x)=2x+2=0⟹x=−1so, minima at x=−1and minimum value of g(−1)=2so range of g(x)=[2,∞]Let g(x)=zso f(x)=z2−4z+6f′(x)=2z−4=0⟹z=4and minimum value f(2)=4for range of f(x)=A=[4,∞]

Let A be range of function $f (x) = (x^{2} + 2 x + 3)^{2} - 4 (x^{2} + 2 x + 3) + 6$ . Then which of the following sets the subsets of A?