English

For what value of λ is the function defined by f(x) = {λ⁢(𝑥2 − 2⁢𝑥), if 𝑥 ≤ 0, 4⁢𝑥 + 1, if 𝑥 > 0 continuous at x = 0? What about continuity at x = 1? - Mathematics

Advertisements
Advertisements

Question

For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if"  x <= 0),(4x+ 1", if"  x > 0):}` continuous at x = 0? What about continuity at x = 1?

Sum
Advertisements

Solution

f(x) = `{(λ(x^2 - 2x)", if"  x <= 0),(4x+ 1", if"  x > 0):}`

If f(x) is continuous at x = 0, it implies:

f(0) = `lim_(x -> 0^+)` f(x) = `lim_(x -> 0^-)` f(x)

⇒ [02 − 2(0)] = [4(0) + 1] = [02 − 2(0)]

⇒ 0 = 1 = 0

Which cannot be true, i.e. for any value of  λ this function is not continuous at x = 0.

If f(x) is continuous at x = 1, this implies:

f(1) = `lim_(x -> 1^+)` f (x) = `lim_(x -> 1^-)` f(x)

⇒ 4(1) + 1 = 4(1) + 1 = 4(1) + 1

⇒ 5 = 5 = 5

Which is always true, i.e. for any value of  λ this function is continuous at x = 1.

shaalaa.com
  Is there an error in this question or solution?
Chapter 5: Continuity and Differentiability - Exercise 5.1 [Page 160]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 5 Continuity and Differentiability
Exercise 5.1 | Q 18 | Page 160

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.


Discuss the continuity of the cosine, cosecant, secant and cotangent functions.


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

f(x) = `{((kcosx)/(pi-2x)", if"  x != pi/2),(3", if"  x = pi/2):}` at x = `"pi/2`


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

f(x) = `{(kx + 1", if"  x <= 5),(3x - 5", if"  x > 5):}` at x = 5


Examine that sin |x| is a continuous function.


Find the values of a so that the function 

\[f\left( x \right) = \begin{cases}ax + 5, if & x \leq 2 \\ x - 1 , if & x > 2\end{cases}\text{is continuous at x} = 2 .\]

Find the value of k if f(x) is continuous at x = π/2, where \[f\left( x \right) = \begin{cases}\frac{k \cos x}{\pi - 2x}, & x \neq \pi/2 \\ 3 , & x = \pi/2\end{cases}\]


Let  \[f\left( x \right) = \frac{\log\left( 1 + \frac{x}{a} \right) - \log\left( 1 - \frac{x}{b} \right)}{x}\] x ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0.

 


If  \[f\left( x \right) = \frac{2x + 3\ \text{ sin }x}{3x + 2\ \text{ sin }  x}, x \neq 0\] If f(x) is continuous at x = 0, then find f (0).


In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}(x - 1)\tan\frac{\pi  x}{2}, \text{ if } & x \neq 1 \\ k , if & x = 1\end{cases}\] at x = 1at x = 1


Discuss the continuity of the function  

\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x}{\left| x \right|}, & x \neq 0 \\ 0 , & x = 0\end{array} . \right.\]

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:  \[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, & \text{ if }  x \neq 0 \\ 3k , & \text{ if  } x = 0\end{cases}\] 


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


In the following, determine the value of constant involved in the definition so that the given function is continuou:   \[f\left( x \right) = \begin{cases}4 , & \text{ if } x \leq - 1 \\ a x^2 + b, & \text{ if }  - 1 < x < 0 \\ \cos x, &\text{ if }x \geq 0\end{cases}\]


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


What happens to a function f (x) at x = a, if  

\[\lim_{x \to a}\] f (x) = f (a)?

If \[f\left( x \right) = \begin{cases}\frac{x}{\sin 3x}, & x \neq 0 \\ k , & x = 0\end{cases}\]  is continuous at x = 0, then write the value of k.


If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to 


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 


Let  \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}, x \neq \frac{\pi}{4} .\]  The value which should be assigned to f (x) at  \[x = \frac{\pi}{4},\]so that it is continuous everywhere 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 =

 


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


Let f (x) = |cos x|. Then,


Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if


If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\] 

then at x = 0, f (x) is


The function f(x) = `"e"^|x|` is ______.


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


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

`lim_("x" -> 0) (1 - "cos x")/"x sin x"` is equal to ____________.


The function f(x) = x2 – sin x + 5 is continuous at x =


For what value of `k` the following function is continuous at the indicated point

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


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×