English

The Function F (X) = 1 + |Cos X| is - Mathematics

Advertisements
Advertisements

Question

The function f (x) = 1 + |cos x| is

Options

  • continuous no where

  • continuous everywhere

  •  not differentiable at x = 0

  • not differentiable at x = n π, n ∈ Z

MCQ
Answer in Brief

Solution

(b) continuous everywhere
Graph of the function f (x) = 1 + |cos x| is as shown below:

From the graph, we can see that f (x) is everywhere continuous but not differentiable at

\[x = \left( 2n + 1 \right)\frac{\pi}{2}, n \in Z\]
shaalaa.com
  Is there an error in this question or solution?
Chapter 10: Differentiability - Exercise 10.4 [Page 19]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 10 Differentiability
Exercise 10.4 | Q 18 | Page 19

RELATED QUESTIONS

Find the relationship between a and b so that the function f defined by `f(x)= {(ax + 1, if x<= 3),(bx + 3, if x  > 3):}` is continuous at x = 3.


Find the values of k so that the function f is continuous at the indicated point.

`f(x) = {(kx^2, "," if x<= 2),(3, "," if x > 2):} " at x" = 2`


Show that the function defined by f (x) = cos (x2) is a continuous function.


Show that the function defined by f(x) = |cos x| is a continuous function.


Determine the value of the constant k so that the function

\[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0 .\]

 


If  \[f\left( x \right) = \begin{cases}2 x^2 + k, &\text{ if }  x \geq 0 \\ - 2 x^2 + k, & \text{ if }  x < 0\end{cases}\]  then what should be the value of k so that f(x) is continuous at x = 0.

 


Find the points of discontinuity, if any, of the following functions: 

\[f\left( x \right) = \begin{cases}x^3 - x^2 + 2x - 2, & \text{ if }x \neq 1 \\ 4 , & \text{ if } x = 1\end{cases}\]

 


Find the points of discontinuity, if any, of the following functions:  \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & \text{ if }  x < 0 \\ 2x + 3, & x \geq 0\end{cases}\]


In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}kx + 5, & \text{ if  }  x \leq 2 \\ x - 1, & \text{ if }  x > 2\end{cases}\]


In the following, determine the value of constant involved in the definition so that the given function is continuou:  \[f\left( x \right) = \begin{cases}k( x^2 + 3x), & \text{ if }  x < 0 \\ \cos 2x , & \text{ if }  x \geq 0\end{cases}\]


Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x


Show that f (x) = | cos x | is a continuous function.

 

If  \[f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}\]  is continuous at x = 0, find k


If \[f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]is continuous at x = 0, write the value of k.


Determine the value of the constant 'k' so that function 

\[\left( x \right) = \begin{cases}\frac{kx}{\left| x \right|}, &\text{ if }  x < 0 \\ 3 , & \text{ if } x \geq 0\end{cases}\]  is continuous at x  = 0  . 

If  \[f\left( x \right) = \begin{cases}\frac{\log\left( 1 + ax \right) - \log\left( 1 - bx \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] and f (x) is continuous at x = 0, then the value of k is


The value of f (0), so that the function

\[f\left( x \right) = \frac{\left( 27 - 2x \right)^{1/3} - 3}{9 - 3 \left( 243 + 5x \right)^{1/5}}\left( x \neq 0 \right)\] is continuous, is given by 


If  \[f\left( x \right) = \frac{1 - \sin x}{\left( \pi - 2x \right)^2},\] when x ≠ π/2 and f (π/2) = λ, then f (x) will be continuous function at x= π/2, where λ =


The function 

\[f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, & x \neq 0 \\ \frac{k}{2} , & x = 0\end{cases}\]  is continuous at x = 0, then k =

If the function  \[f\left( x \right) = \frac{2x - \sin^{- 1} x}{2x + \tan^{- 1} x}\] is continuous at each point of its domain, then the value of f (0) is 


If the function f (x) defined by  \[f\left( x \right) = \begin{cases}\frac{\log \left( 1 + 3x \right) - \log \left( 1 - 2x \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then k =

 


Find the values of a and b so that the function

\[f\left( x \right)\begin{cases}x^2 + 3x + a, & \text { if } x \leq 1 \\ bx + 2 , &\text {  if } x > 1\end{cases}\] is differentiable at each x ∈ R.

Find the values of a and b, if the function f defined by 

\[f\left( x \right) = \begin{cases}x^2 + 3x + a & , & x \leqslant 1 \\ bx + 2 & , & x > 1\end{cases}\] is differentiable at = 1.

If is defined by  \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\] 


The function f (x) = x − [x], where [⋅] denotes the greatest integer function is


The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is


The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.


If f.g is continuous at x = a, then f and g are separately continuous at x = a.


`lim_("x" -> 0) ("x cos x" - "log" (1 + "x"))/"x"^2` is equal to ____________.


The point(s), at which the function f given by f(x) = `{("x"/|"x"|","  "x" < 0),(-1","  "x" ≥ 0):}` is continuous, is/are:


Find the values of `a` and ` b` such that the function by:

`f(x) = {{:(5",", if  x ≤ 2),(ax + b",", if 2 < x < 10),(21",", if x ≥ 10):}`

is a continuous function.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×