Let fx be a continuous and increasing function on - and suppose Fx - 1 - fx - f2x - f3x-If the equation Fx - 0 has a positive root- then-

We have, F(x) = 1 + f(x) + f^2(x) + f^3(x) ⇒ F(x) = (1 + f(x)) (1 + f^2 (x)) F(x) = 0 nbsp;⇒ nbsp; f(x) = 1 nbsp; f( α ) = 1 nbsp; where α ϵ ℝ^+ So α ...

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Standard XI

Mathematics

Monotonically Increasing Functions

Let fx be a c...

Question

Let $f (x)$ be a continuous and increasing function on $R$ and suppose $F (x) = 1 + f (x) + f^{2} (x) + f^{3} (x) .$
If the equation $F (x) = 0$ has a positive root, then:

F (x)

is a decreasing function of

x

R

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f (0) < - 1

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f (0) > 0

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| f (0) | < 1

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Solution

The correct option is B $f (0) < - 1$
We have,
$F (x) = 1 + f (x) + f^{2} (x) + f^{3} (x)$
$\Rightarrow F (x) = (1 + f (x)) (1 + f^{2} (x))$
$F (x) = 0 \Rightarrow f (x) = - 1$
$f (α) = - 1$ where $α ϵ R^{+}$
So $α > 0$ and as $f (x)$ is an increasing function.
$f (α) > f (0)$
So $f (0) < - 1$

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Let f(x) be a continuous and increasing function on R and suppose F(x)=1+f(x)+f2(x)+f3(x). If the equation F(x)=0 has a positive root, then:

The correct option is B f(0)<−1We have, F(x)=1+f(x)+f2(x)+f3(x) ⇒F(x)=(1+f(x))(1+f2(x)) F(x)=0⇒f(x)=−1 f(α)=−1 where α ϵ R+ So α>0 and as f(x) is an increasing function. f(α)>f(0) So f(0)<−1

Let $f (x)$ be a continuous and increasing function on $R$ and suppose $F (x) = 1 + f (x) + f^{2} (x) + f^{3} (x) .$
If the equation $F (x) = 0$ has a positive root, then:

The correct option is B $f (0) < - 1$
We have,
$F (x) = 1 + f (x) + f^{2} (x) + f^{3} (x)$
$\Rightarrow F (x) = (1 + f (x)) (1 + f^{2} (x))$
$F (x) = 0 \Rightarrow f (x) = - 1$
$f (α) = - 1$ where $α ϵ R^{+}$
So $α > 0$ and as $f (x)$ is an increasing function.
$f (α) > f (0)$
So $f (0) < - 1$