First slide

Monotonicity

Question

If $ϕ (x)$ is a polynomial function and $ϕ^{'} (x) > ϕ (x) \forall x \geq 1$ and $ϕ (1) = 0,$ then

Moderate

A

$ϕ (x) \geq 0 \forall x \geq 1$
B

$ϕ (x) < 0 \forall x \geq 1$
C

$ϕ (x) = 0 \forall x \geq 1$
D

none of these

Solution

Given $ϕ^{'} (x) - ϕ (x) > 0 \forall x \geq 1$
or $e^{- x} \{ϕ^{'} (x) - ϕ (x)\} > 0 \forall x \geq 1$
or $\frac{d}{dx} e^{- x} ϕ (x) > 0 \forall x \geq 1$
Therefore, $e^{- x} ϕ (x)$ ) is an increasing function $\forall x \geq 1$ .
Since $ϕ (x)$ is a polynomial,
$e^{- x} ϕ (x) > e^{- 1} ϕ (1) or e^{- x} ϕ (x) > 0 [∵ ϕ (1) = 0]$
or $ϕ (x) > 0$

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