Rolle's theorem is applicable in the interval $[-2, 2]$ for the function
- $f \left(x\right)=x^{3}$
- $f \left(x\right)=4X^{4}$
- $f \left(x\right)=2X^{3}+3$
- $f \left(x\right)=\pi\left|x\right|$
The Correct Option is B
Solution and Explanation
(i) $f(x)$ is continuous in $(-2,2)$
(ii) $f(x)$ is differentiable in $(-2,2)$
(iii) $f(-2)=f(2)$
So, $f(x)=4 x^{4}$ satisfies all the conditions of Rolle's theorem therefore $\exists$ a point $c$ such that $f'(c)=0$
$\Rightarrow 16 c^{3}=0 \Rightarrow c=0 \in(-2,2)$
Top Questions on Differentiability
Match the following:
In the following, \( [x] \) denotes the greatest integer less than or equal to \( x \).
Choose the correct answer from the options given below:- KCET - 2025
- Mathematics
- Differentiability
- The function f(x) is given by:
For x < 0:
f(x) = ex + axFor x ≥ 0:
f(x) = b(x - 1)2
The function is differentiable at x = 0. Then,- KCET - 2025
- Mathematics
- Differentiability
- If \[ y = \frac{\cos x}{1 + \sin x} \] then:
- KCET - 2025
- Mathematics
- Differentiability
- The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is.
- JEE Main - 2025
- Mathematics
- Differentiability
Let the function, \(f(x)\) = \(\begin{cases} -3ax^2 - 2, & x < 1 \\a^2 + bx, & x \geq 1 \end{cases}\) Be differentiable for all \( x \in \mathbb{R} \), where \( a > 1 \), \( b \in \mathbb{R} \). If the area of the region enclosed by \( y = f(x) \) and the line \( y = -20 \) is \( \alpha + \beta\sqrt{3} \), where \( \alpha, \beta \in \mathbb{Z} \), then the value of \( \alpha + \beta \) is:
- JEE Main - 2025
- Mathematics
- Differentiability
Questions Asked in WBJEE exam
- If 1000! = 3n × m, where m is an integer not divisible by 3, then n =?
- WBJEE - 2024
- Probability
- Chords AB CD of a circle intersect at right angle at the point P. If the lengths of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is:
- If the relation between the direction ratios of two lines in \(\mathbb{R}^3\) are given by \(l + m + n = 0\), \(2lm + 2mn - ln = 0\), then the angle between the lines is:
- WBJEE - 2024
- Vectors
- If \(a, b, c\) are distinct odd natural numbers, then the number of rational roots of the equation \(ax^2 + bx + c = 0\) is:
- WBJEE - 2024
- Quadratic Equation
- What force \(F\) is required to start moving this 10 kg block shown in the figure if it acts at an angle of \(60^\circ\) as shown? (\(\mu_s = 0.6\)).
- WBJEE - 2024
- laws of motion
Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.