Let fx - ax2 - 2a-1 x-9a-4x2-8x-32 and gx - x-x2 -1- If fx is less than M - 34 for all x - where M is the maximum value of gx - then set of values of a lies in

Let $f (x) = x^{2}$ and $g (x) = sin x$ for all $x \in R$ . Then the set of all $x$ satifying $(f \circ g \circ g \circ f) (x) = (g \circ g \circ f) (x)$ , where $(f \circ g) (x) = f (g (x))$ , is

Q. Find the range of values of

a

, such that

f (x) = \frac{a x^{2} + 2 (a + 1) x + 9 a + 4}{x^{2} - 8 x + 32}

is always negative.

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Let f(x)=ax2+2(a+1)x+9a+4x2−8x+32 and g(x)=x−x2−1. If f(x) is less than M+34 for all x∈R, where M is the maximum value of g(x), then set of values of a lies in

The correct option is D (−∞,0)M=−(1−4(−1)(−1))4(−1)=−34 ∴f(x)<0 ∀ x∈R ⇒ax2+2(a+1)x+9a+4<0 ∀ x ∴a<0 and D<0 D=4(a+1)2−4a(9a+4)<0⇒a2+2a+1−9a2−4a<0⇒−8a2−2a+1<0⇒8a2+2a−1>0⇒8a2+4a−2a−1>0⇒(2a+1)(4a−1)>0 But a<0 ∴a∈(−∞,−12)

Let $f (x) = \frac{a x^{2} + 2 (a + 1) x + 9 a + 4}{x^{2} - 8 x + 32}$ and $g (x) = x - x^{2} - 1.$ If $f (x)$ is less than $M + \frac{3}{4}$ for all $x \in R,$ where $M$ is the maximum value of $g (x),$ then set of values of $a$ lies in