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Question:
The equation $x log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$.
The function $f(x) = x log x$ is an increasing function in $[1,2]$ and $g (x)=2-x$ is a decreasing function in $[1, 2]$ and the graphs represented by these functions intersect at a point in $[1,2]$
The equation $x log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$.
The function $f(x) = x log x$ is an increasing function in $[1,2]$ and $g (x)=2-x$ is a decreasing function in $[1, 2]$ and the graphs represented by these functions intersect at a point in $[1,2]$
Updated On: Aug 21, 2024
- Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- Statement-1 is true; Statement-2 is true; Statement-2 is not correct explanation for Statement-1.
- Statement-1 is false, Statement-2 is true.
- Statement-1 is true, Statement-2 is false.
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The Correct Option is A
Solution and Explanation
$f\left(x\right)=x\,log\,x,\,f\left(1\right) = 0,\,f\left(2\right) = 4$
$g\left(x\right) = 2-x,\,g\left(1\right)=1,\,g\left(2\right) = 0$
$log 10 > log\,4 \Rightarrow 1 > log\,4$
Thus statement -1 and 2 both are true and statement-2 is a correct explanation of statement 1.
$g\left(x\right) = 2-x,\,g\left(1\right)=1,\,g\left(2\right) = 0$
$log 10 > log\,4 \Rightarrow 1 > log\,4$
Thus statement -1 and 2 both are true and statement-2 is a correct explanation of statement 1.
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Concepts Used:
Increasing and Decreasing Functions
Increasing Function:
On an interval I, a function f(x) is said to be increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≤ f(y)
Decreasing Function:
On an interval I, a function f(x) is said to be decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≥ f(y)
Strictly Increasing Function:
On an interval I, a function f(x) is said to be strictly increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) < f(y)
Strictly Decreasing Function:
On an interval I, a function f(x) is said to be strictly decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) > f(y)
Graphical Representation of Increasing and Decreasing Functions
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