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Application Of Derivatives

Question

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Show that the function f (x) = x² is an increasing function in (0, ∞).

Solution

Let x₁ , x₂ ∊ (0, ∞) and let x₁, < x₂.
Now, $straight x subscript 1 space less than space straight x subscript 2$
$rightwards double arrow space space space space space straight x subscript 1. space straight x subscript 1 space space space less than space straight x subscript 1. space straight x subscript 2$ $open square brackets because space space straight x subscript 1 greater than 0 close square brackets$

rightwards double arrow space space space space straight x subscript 1 squared space space less than space straight x subscript 1. straight x subscript 2

...(1)
Again

straight x subscript 1 less than straight x subscript 2

rightwards double arrow space space space space space straight x subscript 1. space space straight x subscript 2 space space space less than space space straight x subscript 2. space straight x subscript 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space

open square brackets because space space straight x subscript 2 space greater than space 0 close square brackets

rightwards double arrow space space space straight x subscript 1. space end subscript straight x subscript 2 space space less than space straight x subscript 2 squared

...(2)
From (1) and (2), we get,

straight x subscript 1 squared space less than space straight x subscript 2 squared space space space space space space or space space space space space straight f left parenthesis straight x subscript 1 right parenthesis space less than space straight f left parenthesis straight x subscript 2 right parenthesis

therefore space space straight x subscript 1 space less than space straight x subscript 2 space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight f left parenthesis straight x subscript 1 right parenthesis space less than space straight f left parenthesis straight x subscript 2 right parenthesis therefore space space space space straight f space is space an space increasing space function space in space left parenthesis 0 comma space infinity right parenthesis.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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