Let f, g and h are differentiable function such that g(x)=f(x)-x and h(x)=f(x)-x3 are both strictly increasing functions, then the function F(x)=f(x)-(√3 x2/2) is

Question

Let f, g and h are differentiable function such that g(x)=f(x)-x and h(x)=f(x)-x3 are both strictly increasing functions, then the function F(x)=f(x)-(√3 x2/2) is (A) strictly increasing ∀ x ∈ R (B) strictly decreasing ∀ x ∈ R (C) strictly decreasing on (-∞, (1/√3)) and strictly increasing on ((1/√3), ∞) (D) strictly increasing on (-∞, (1/√3)) and strictly decreasing on ((1/√3), ∞)

Tardigrade Admin · Accepted Answer

Assume f ( x )= x 3+ x so that g ( x )= x 3 ; h ( x )= x and F ( x )= x 3-(√3 x 2/2)+ x F prime( x )=3 x 2-√3 x +1 D =3-12<0 F prime( x )>0 ∀ x ∈ R ⇒ F is increasing ⇒ (A)

Q. Let $f, g$ and $h$ are differentiable function such that $g (x) = f (x) - x$ and $h (x) = f (x) - x^{3}$ are both strictly increasing functions, then the function $F (x) = f (x) - \frac{3 x ^{2}}{2}$ is

strictly increasing $\forall x \in R$

strictly decreasing $\forall x \in R$

strictly decreasing on $(- \infty, \frac{1}{3})$ and strictly increasing on $(\frac{1}{3}, \infty)$

strictly increasing on $(- \infty, \frac{1}{3})$ and strictly decreasing on $(\frac{1}{3}, \infty)$

Assume $f (x) = x^{3} + x$ so that $g (x) = x^{3}; h (x) = x$ and $F (x) = x^{3} - \frac{3 x ^{2}}{2} + x$ $F^{'} (x) = 3 x^{2} - 3 x + 1$
$D = 3 - 12 < 0$
$F^{'} (x) > 0\forall x \in R \Rightarrow F$ is increasing $\Rightarrow$ (A)

Q. Let f,g and h are differentiable function such that g(x)=f(x)−x and h(x)=f(x)−x3 are both strictly increasing functions, then the function F(x)=f(x)−23​x2​ is

strictly increasing ∀x∈R

strictly decreasing ∀x∈R

strictly decreasing on (−∞,3​1​) and strictly increasing on (3​1​,∞)

strictly increasing on (−∞,3​1​) and strictly decreasing on (3​1​,∞)