Question

Let f and g be functions from the interval $[0, \infty)$ to the interval $[0, \infty)$,f being an increasing and g being a decreasing function. If f{g(0)} = 0 then

• A. f{g(x)} $\ge$ f{g(0)}
• B. g{f (x)} $\le$ g{f (0)}
• C. f {g(2)} = 7
• D. none of these

Answer 1

Answer: (B)

f '(x) > 0 if x $\ge$ 0 and g'(x) < 0 if x $\ge$ 0
Let h(x) = f (g(x))
then h'(x) = f '(g(x)).g'(x) < 0 if x $\ge$ 0
$\therefore $ h(x) is decreasing function
$\therefore $ h(x) $\le$ h(0) if x $\ge$ 0
$\therefore $ f (g(x)) $\le$ f (g(0)) = 0
But codomain of each function is [0, $\infty$ )
$\therefore $ f (g(x)) = 0 for all x $\ge$ 0
$\therefore $ f (g(x)) = 0
Also g( f (x)) $\le$ g( f (0)) [as above]