First slide

Monotonicity

Question

$Let f : R \to R, f (0) = 0, f^{'} (x) > 0 and f^{''} (x) > 0 and A (α, f (α)), B (β, f (β)), C (γ, f (γ))$ $be three points on the graph of y = f (x) and$
$0 < α < β < γ . Then which of the following is true?$

Moderate

A

$γ f (γ + β - α) > (γ + β - α) f (γ)$
B

$γ f (γ + β - α) < (γ + β - α) f (γ)$
C

$α f (γ + β - α) < (γ + β - α) f (α)$
D

$α f (β) < β f (α)$

Solution

$\begin{array}{l} \frac{f (x)}{x} is an increasing function (s l o p e o f O P = \frac{f (x) - f (0)}{x - 0} = \frac{f (x)}{x} i s i n c r e a \sin g, w h e r e P (x, f (x))) \\ ∴ \frac{f (γ + β - α)}{γ + β - α} > \frac{f (γ)}{γ} (\sin c e β - α > 0 \Rightarrow γ + β - α > γ) \\ \Rightarrow γ f (γ + β - α) > (γ + β - α) f (γ) \end{array}$

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