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

Updated On: Jul 6, 2022
  • f{g(x)} $\ge$ f{g(0)}
  • g{f (x)} $\le$ g{f (0)}
  • f {g(2)} = 7
  • none of these
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The Correct Option is B

Solution and Explanation

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]
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Concepts Used:

Application of Derivatives

Various Applications of Derivatives-

Rate of Change of Quantities:

If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by 

\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)

This is also known to be as the Average Rate of Change.

Increasing and Decreasing Function:

Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).

  • If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
  • Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
  • The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.

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