English

Find the values of a and b such that the function defined by f(x) = {5, if ๐‘ฅ โ‰ค 2, ๐‘Žโข๐‘ฅ + ๐‘, if 2 < ๐‘ฅ < 10, 21, if ๐‘ฅ โ‰ฅ 10 is a continuous function. - Mathematics

Advertisements
Advertisements

Question

Find the values of a and b such that the function defined by f(x) = `{(5", if"  x <= 2),(ax +b", if"  2 < x < 10),(21", if"  x >= 10):}` is a continuous function.

Sum
Advertisements

Solution

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

Since f(x) = 5, f(x) = ax + b, f(x) is a continuous function at 21 times, f(x) is already a continuous function at x < 2, 2 < x < 10, x > 10.

If f(x) is continuous at x = 2, this implies:

f(2) = `lim_(x -> 2^+)` f(x) = `lim_(x -> 2^-)` f(x)

⇒ 5 = a(2) + b 

⇒ 2a + b = 5   ...(1)

If f(x) is continuous at x = 10, this implies:

f(10) = `lim_(x -> 10^+)` f(x) = `lim_(x -> 10^-)` f(x)

⇒ 21 = a(10) + b

⇒ 10a + b  = 21   ...(2)

Subtracting equation (2) from (1),

⇒ 8a = 16

⇒ a = `16/8`

⇒ a = 2

Put a = 2 in equation (1)

⇒ 2(2) + b  = 5

⇒ 4 + b = 5

⇒ b = 5 − 4

⇒ b = 1

That is, the function f(x) is continuous for the quantities a = 2, b = 1.

shaalaa.com
  Is there an error in this question or solution?
Chapter 5: Continuity and Differentiability - Exercise 5.1 [Page 161]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 5 Continuity and Differentiability
Exercise 5.1 | Q 30 | Page 161

RELATED QUESTIONS

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?


Is the function defined by f(x) = x2 − sin x + 5 continuous at x = π?


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


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


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


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


Find the values of a and b so that the function f given by \[f\left( x \right) = \begin{cases}1 , & \text{ if } x \leq 3 \\ ax + b , & \text{ if } 3 < x < 5 \\ 7 , & \text{ if }  x \geq 5\end{cases}\] is continuous at x = 3 and x = 5.


If \[f\left( x \right) = \begin{cases}\frac{x^2}{2}, & \text{ if } 0 \leq x \leq 1 \\ 2 x^2 - 3x + \frac{3}{2}, & \text P{ \text{ if }  }  1 < x \leq 2\end{cases}\]. Show that f is continuous at x = 1.

 

Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 - 16}{x - 2}, & \text{ if } x \neq 2 \\ 16 , & \text{ if }  x = 2\end{cases}\]


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:  \[f\left( x \right) = \begin{cases}k( x^2 + 3x), & \text{ if }  x < 0 \\ \cos 2x , & \text{ if }  x \geq 0\end{cases}\]


The function  \[f\left( x \right) = \begin{cases}x^2 /a , & \text{ if } 0 \leq x < 1 \\ a , & \text{ if } 1 \leq x < \sqrt{2} \\ \frac{2 b^2 - 4b}{x^2}, & \text{ if }  \sqrt{2} \leq x < \infty\end{cases}\] is continuous on (0, ∞), then find the most suitable values of a and b.


The function f(x) is defined as follows: 

\[f\left( x \right) = \begin{cases}x^2 + ax + b , & 0 \leq x < 2 \\ 3x + 2 , & 2 \leq x \leq 4 \\ 2ax + 5b , & 4 < x \leq 8\end{cases}\]

If f is continuous on [0, 8], find the values of a and b.


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 the function   \[f\left( x \right) = \frac{\sin 10x}{x}, x \neq 0\] is continuous at x = 0, find f (0).

 


Determine whether \[f\left( x \right) = \binom{\frac{\sin x^2}{x}, x \neq 0}{0, x = 0}\]  is continuous at x = 0 or not.

 


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.


 then f (x) is continuous for all
\[f\left( x \right) = \begin{cases}\frac{\left| x^2 - x \right|}{x^2 - x}, & x \neq 0, 1 \\ 1 , & x = 0 \\ - 1 , & x = 1\end{cases}\]  then f (x) is continuous for all

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 


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 =

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.

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


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


Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.


The function f(x) = x |x| is ______.


Share
Notifications

Englishเคนเคฟเค‚เคฆเฅ€เคฎเคฐเคพเค เฅ€


      Forgot password?
Use app×