मराठी

If F (X) = | X − a | ϕ (X), Where ϕ (X) is Continuous Function, Then (A) F' (A+) = ϕ (A) (B) F' (A−) = −ϕ (A) (C) F' (A+) = F' (A−) (D) None of These - Mathematics

Advertisements
Advertisements

प्रश्न

If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then

पर्याय

  • f' (a+) = ϕ (a)

  • f' (a) = −ϕ (a)

  • f' (a+) = f' (a)

  • none of these

MCQ

उत्तर

\[f'\left( a^+ \right) = \phi\left( a \right)\]

\[f'\left( a^- \right) = - \phi\left( a \right)\]

Here, 

\[f\left( x \right) = \left| x - a \right| \phi\left( x \right)\]
\[f'\left( a^+ \right) = \lim_{h \to 0} \frac{f\left( a + h \right) - f\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| h + a - a \right|\phi\left( a + h \right) - \left| a - a \right|\phi\left( a \right)}{h} = \lim_{h \to 0} \frac{h \phi\left( a + h \right)}{h} = \lim_{h \to 0} \phi\left( a + h \right) = \phi\left( a \right)\]

Also,

\[f'\left( a^- \right) = \lim_{h \to 0} \frac{f\left( a - h \right) - f\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| a - h - a \right| \phi\left( a - h \right) - \left| a - a \right| \phi\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| - h \right| \phi\left( a - h \right)}{h} = \lim_{h \to 0} - \phi\left( a - h \right) = - \phi\left( a \right)\]
shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 9: Continuity - Exercise 9.4 [पृष्ठ ४२]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12
पाठ 9 Continuity
Exercise 9.4 | Q 2 | पृष्ठ ४२

व्हिडिओ ट्यूटोरियलVIEW ALL [4]

संबंधित प्रश्‍न

 If 'f' is continuous at x = 0, then find f(0).

`f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0`


Determine the value of 'k' for which the following function is continuous at x = 3

`f(x) = {(((x + 3)^2 - 36)/(x - 3),  x != 3), (k,  x = 3):}`


If \[f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\]

Find whether f(x) is continuous at x = 0.

 

Discuss the continuity of the following functions at the indicated point(s): (iv) \[f\left( x \right) = \left\{ \begin{array}{l}\frac{e^x - 1}{\log(1 + 2x)}, if & x \neq a \\ 7 , if & x = 0\end{array}at x = 0 \right.\]


If  \[f\left( x \right) = \begin{cases}\frac{1 - \cos kx}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\text{is continuous at x} = 0, \text{ find } k .\]


In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  

\[f\left( x \right) = \begin{cases}k( x^2 - 2x), \text{ if }  & x < 0 \\ \cos x, \text{ if }  & x \geq 0\end{cases}\] at x = 0

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; 

\[f\left( x \right) = \begin{cases}kx + 1, \text{ if }  & x \leq \pi \\ \cos x, \text{ if }  & x > \pi\end{cases}\] at x = π

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  \[f\left( x \right) = \begin{cases}kx + 1, if & x \leq 5 \\ 3x - 5, if & x > 5\end{cases}\] at x = 5


Let\[f\left( x \right) = \left\{ \begin{array}\frac{1 - \sin^3 x}{3 \cos^2 x} , & \text{ if }  x < \frac{\pi}{2} \\ a , & \text{ if }  x = \frac{\pi}{2} \\ \frac{b(1 - \sin x)}{(\pi - 2x )^2}, & \text{ if }  x > \frac{\pi}{2}\end{array} . \right.\] ]If f(x) is continuous at x = \[\frac{\pi}{2}\] , find a and b.

 

Prove that
\[f\left( x \right) = \begin{cases}\frac{\sin x}{x} , & x < 0 \\ x + 1 , & x \geq 0\end{cases}\] is everywhere continuous.

 


Find all the points of discontinuity of f defined by f (x) = | x |− | x + 1 |.


Determine if \[f\left( x \right) = \begin{cases}x^2 \sin\frac{1}{x} , & x \neq 0 \\ 0 , & x = 0\end{cases}\] is a continuous function?

 


The function 

\[f\left( x \right) = \frac{4 - x^2}{4x - x^3}\]

 


The value of b for which the function 

\[f\left( x \right) = \begin{cases}5x - 4 , & 0 < x \leq 1 \\ 4 x^2 + 3bx , & 1 < x < 2\end{cases}\] is continuous at every point of its domain, is 

If  \[f\left( x \right) = \left\{ \begin{array}a x^2 + b , & 0 \leq x < 1 \\ 4 , & x = 1 \\ x + 3 , & 1 < x \leq 2\end{array}, \right.\] then the value of (ab) for which f (x) cannot be continuous at x = 1, is

 


If  \[f\left( x \right) = \begin{cases}\frac{1 - \sin^2 x}{3 \cos^2 x} , & x < \frac{\pi}{2} \\ a , & x = \frac{\pi}{2} \\ \frac{b\left( 1 - \sin x \right)}{\left( \pi - 2x \right)^2}, & x > \frac{\pi}{2}\end{cases}\]. Then, f (x) is continuous at  \[x = \frac{\pi}{2}\], if

 


Show that f(x) = x1/3 is not differentiable at x = 0.


Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat: 

\[f\left( x \right) = \begin{cases}3x - 2, & 0 < x \leq 1 \\ 2 x^2 - x, & 1 < x \leq 2 \\ 5x - 4, & x > 2\end{cases}\]

Write the points of non-differentiability of 

\[f \left( x \right) = \left| \log \left| x \right| \right| .\]

The function f (x) = e|x| is


If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]


Examine the continuity of f(x)=`x^2-x+9  "for"  x<=3`

=`4x+3  "for"  x>3,  "at"  x=3` 


Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4 

= 10 for x = 4  at x = 4


Discuss the continuity of the function f at x = 0

If f(x) = `(2^(3x) - 1)/tanx`, for x ≠ 0

         = 1,   for x = 0


Find the points of discontinuity , if any for the function : f(x) = `(x^2 - 9)/(sinx - 9)`


If the function f is continuous at x = 0

Where f(x) = 2`sqrt(x^3 + 1)` + a,  for x < 0,
= `x^3 + a + b,  for x > 0
and f (1) = 2, then find a and b.


If f(x) = `(e^(2x) - 1)/(ax)` .                for x < 0 , a ≠ 0
         = 1.                             for x = 0
         = `(log(1 + 7x))/(bx)`.        for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b


Find the value of 'k' if the function 
f(x) = `(tan 7x)/(2x)`,                   for x ≠ 0.
      = k                                        for x = 0.
is continuous at x = 0.


Examine the continuity of the followin function : 

  `{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`   


Show that the function f defined by f(x) = `{{:(x sin  1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.


The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.


The function given by f (x) = tanx is discontinuous on the set ______.


Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1


f(x) = `{{:(|x|cos  1/x",", "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0


Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))


Examine the differentiability of f, where f is defined by
f(x) = `{{:(x^2 sin  1/x",",  "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0


A function f: R → R satisfies the equation f( x + y) = f(x) f(y) for all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f′(0) = 2. Prove that f′(x) = 2f(x).


Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",",  "if"  x ≤ 1),("q"x + 2",",  "if"  x > 1):}` is differentiable at x = 1


The value of k (k < 0) for which the function f defined as

f(x) = `{((1-cos"kx")/("x"sin"x")","  "x" ≠ 0),(1/2","  "x" = 0):}`

is continuous at x = 0 is:


If the following function is continuous at x = 2 then the value of k will be ______.

f(x) = `{{:(2x + 1",", if x < 2),(                 k",", if x = 2),(3x - 1",", if x > 2):}`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×