Let F: R arrow R be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F prime(x)

Question

Let F: R arrow R be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F prime(x) < 0 for all x ∈(1 / 2,3). Let f(x)=x F(x) for all x ∈ R. The correct statement(s) is(are) (A) f prime(1) < 0 (B) f(2) < 0 (C)  f prime(x) ≠ 0 for any x ∈(1,3) (D) f prime(x)=0 for some x ∈(1,3)

Tardigrade Admin · Accepted Answer

f prime(x)=x F prime(x)+F(x) ⇒ f prime(1)=F prime(1)+F(1)=F prime(1)<0 ⇒ (1) f(2)=2 F(2)<0 ⇒ (2) for x ∈(1,3) f prime(x)=x f prime(x)+f(x)<0 ⇒ (3)

Q. Let $F : R \to R$ be a thrice differentiable function. Suppose that $F (1) = 0, F (3) = - 4$ and $F^{'} (x) < 0$ for all $x \in (1/2, 3)$ . Let $f (x) = x F (x)$ for all $x \in R$ .
The correct statement(s) is(are)

$f^{'} (1) < 0$

$f (2) < 0$

$f^{'} (x) \neq = 0$ for any $x \in (1, 3)$

$f^{'} (x) = 0$ for some $x \in (1, 3)$

$f^{'} (x) = x F^{'} (x) + F (x)$
$\Rightarrow f^{'} (1) = F^{'} (1) + F (1) = F^{'} (1) < 0 \Rightarrow$ (1)
$f (2) = 2 F (2) < 0 \Rightarrow (2)$
for $x \in (1, 3) f^{'} (x) = x f^{'} (x) + f (x) < 0 \Rightarrow (3)$

Q. Let F:R→R be a thrice differentiable function. Suppose that F(1)=0,F(3)=−4 and F′(x)<0 for all x∈(1/2,3). Let f(x)=xF(x) for all x∈R. The correct statement(s) is(are)

f′(1)<0

f(2)<0

f′(x)=0 for any x∈(1,3)

f′(x)=0 for some x∈(1,3)