Consider a function f ( x )=(3 x + a / x 2+3) which has greatest value equal to (3/2). The minimum value of f ( x ) is equal to

Question

Consider a function f ( x )=(3 x + a / x 2+3) which has greatest value equal to (3/2). The minimum value of f ( x ) is equal to (A)  tan ((-π/3)) (B)  sin ((-π/6)) (C)  cos ((-π/3)) (D)  cot ((π/2))

Tardigrade Admin · Accepted Answer

y=f(x)=(3 x+3/x2+3) ⇒ x2 y+3 y=3 x+3 ⇒ x2 y-3 x+3 y-3=0 As x ∈ R, so D ≥ 0 ⇒ 9-4 y(3 y-3) ≥ 0 ⇒ 12 y2-12 y-9 ≤ 0 ⇒ 4 y2-4 y-3 ≤ 0 ⇒ 4 y2-6 y+2 y-3 ≤ 0 ⇒(2 y-3)(2 y+1) ≤ 0 Hence y ∈[(-1/2), (3/2)] Hence minimum value of f ( x ) is (-1/2)= sin ((-π/6))

Q. Consider a function $f (x) = \frac{3 x + a}{x ^{2} + 3}$ which has greatest value equal to $\frac{3}{2}$ .
The minimum value of $f (x)$ is equal to

$tan (\frac{- π}{3})$

$sin (\frac{- π}{6})$

$cos (\frac{- π}{3})$

$cot (\frac{π}{2})$

Q. Consider a function f(x)=x2+33x+a​ which has greatest value equal to 23​. The minimum value of f(x) is equal to

tan(3−π​)

sin(6−π​)

cos(3−π​)

cot(2π​)

y=f(x)=x2+33x+3​⇒x2y+3y=3x+3⇒x2y−3x+3y−3=0 As x∈R, so D≥0 ⇒9−4y(3y−3)≥0⇒12y2−12y−9≤0⇒4y2−4y−3≤0 ⇒4y2−6y+2y−3≤0⇒(2y−3)(2y+1)≤0 Hence y∈[2−1​,23​] Hence minimum value of f(x) is 2−1​=sin(6−π​)

Q. Consider a function $f (x) = \frac{3 x + a}{x ^{2} + 3}$ which has greatest value equal to $\frac{3}{2}$ .
The minimum value of $f (x)$ is equal to

$tan (\frac{- π}{3})$

$sin (\frac{- π}{6})$

$cos (\frac{- π}{3})$

$cot (\frac{π}{2})$