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 $[1,2]$
- 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.
The Correct Option is A
Solution and Explanation
$g\left(x\right) = 2-x,\,g\left(1\right)=1,\,g\left(2\right) = 0$
$log 10 > log\,4 \Rightarrow 1 > log\,4$
Thus statement -1 and 2 both are true and statement-2 is a correct explanation of statement 1.
Top Questions on Increasing and Decreasing Functions
- The function \( f(x) = 6x^4 - 3x^2 - 5 \) is increasing in the set:
- KEAM - 2024
- Mathematics
- Increasing and Decreasing Functions
- If $f(x) = x^2 + bx + 1$ is increasing in the interval $[1, 2]$, then the least value of $b$ is:
- CUET (UG) - 2024
- Mathematics
- Increasing and Decreasing Functions
- The function $x^x$, $x > 0$ is strictly increasing at:
- KCET - 2024
- Mathematics
- Increasing and Decreasing Functions
- If $f(x) = x e^{x^{1-x}}$, then $f(x)$ is:
- KCET - 2024
- Mathematics
- Increasing and Decreasing Functions
- Given: \(5f (x) 4f(\frac 1x) = x^2 - 4 \) & \(y = 9f(x) × x^2\) If y is strictly increasing, then find interval of \(x\).
- JEE Main - 2024
- Mathematics
- Increasing and Decreasing Functions
Questions Asked in JEE Main exam
- Find the value of \[ \sin 70^\circ \left( \cot 10^\circ \cot 70^\circ - 1 \right). \]
- JEE Main - 2025
- Miscellaneous
The value of current \( I \) in the electrical circuit as given below, when the potential at \( A \) is equal to the potential at \( B \), will be _____ A.
- JEE Main - 2025
- Digital Circuits
Two light beams fall on a transparent material block at point 1 and 2 with angle \( \theta_1 \) and \( \theta_2 \), respectively, as shown in the figure. After refraction, the beams intersect at point 3 which is exactly on the interface at the other end of the block. Given: the distance between 1 and 2, \( d = \frac{4}{3} \) cm and \( \theta_1 = \theta_2 = \cos^{-1} \left( \frac{n_2}{2n_1} \right) \), where \( n_2 \) is the refractive index of the block and \( n_1 \) is the refractive index of the outside medium, then the thickness of the block is …….. cm.
- JEE Main - 2025
- Units and measurement
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- Two iron solid discs of negligible thickness have radii \( R_1 \) and \( R_2 \) and moment of inertia \( I_1 \) and \( I_2 \), respectively. For \( R_2 = 2R_1 \), the ratio of \( I_1 \) and \( I_2 \) would be \( \frac{1}{x} \), where \( x \) is:
- JEE Main - 2025
- System of Particles & Rotational Motion
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
