Advertisements
Advertisements
Question
Find the values of a and b so that the function f(x) defined by \[f\left( x \right) = \begin{cases}x + a\sqrt{2}\sin x , & \text{ if }0 \leq x < \pi/4 \\ 2x \cot x + b , & \text{ if } \pi/4 \leq x < \pi/2 \\ a \cos 2x - b \sin x, & \text{ if } \pi/2 \leq x \leq \pi\end{cases}\]becomes continuous on [0, π].
Advertisements
Solution
Given: f is continuous on \[\left[ 0, \pi \right]\] .
∴ f is continuous at x = \[\frac{\pi}{4}\] and \[\frac{\pi}{2}\]
At x = \[\frac{\pi}{4}\], we have
\[\lim_{x \to \frac{\pi}{4}^-} f\left( x \right) = \lim_{h \to 0} f\left( \frac{\pi}{4} - h \right) = \lim_{h \to 0} \left[ \left( \frac{\pi}{4} - h \right) + a\sqrt{2}\sin \left( \frac{\pi}{4} - h \right) \right] = \left[ \frac{\pi}{4} + a\sqrt{2} \sin \left( \frac{\pi}{4} \right) \right] = \left[ \frac{\pi}{4} + a \right]\]
\[\lim_{x \to \frac{\pi}{4}^+} f\left( x \right) = \lim_{h \to 0} f\left( \frac{\pi}{4} + h \right) = \lim_{h \to 0} \left[ 2\left( \frac{\pi}{4} + h \right) \cot \left( \frac{\pi}{4} + h \right) + b \right] = \left[ \frac{\pi}{2} \cot \left( \frac{\pi}{4} \right) + b \right] = \left[ \frac{\pi}{2} + b \right]\]
At x = \[\frac{\pi}{2}\] , we have
\[ \Rightarrow b = \frac{- a}{2} . . . \left( 1 \right) \text{ and } \frac{- \pi}{4} = b - a . . . \left( 2 \right)\]
\[ \Rightarrow \frac{- \pi}{4} = \frac{- 3a}{2} \left[ \text{ Substituting the value of b in eq .} \left( 2 \right) \right]\]
\[ \Rightarrow a = \frac{\pi}{6}\]
\[ \Rightarrow b = \frac{- \pi}{12} \left[ \text{ From eq } . \left( 1 \right) \right]\]
APPEARS IN
RELATED QUESTIONS
If f(x)= `{((sin(a+1)x+2sinx)/x,x<0),(2,x=0),((sqrt(1+bx)-1)/x,x>0):}`
is continuous at x = 0, then find the values of a and b.
Examine the following function for continuity:
f(x) = `1/(x - 5)`, x ≠ 5
A function f(x) is defined as
Show that f(x) is continuous at x = 3
Show that
\[f\left( x \right) = \begin{cases}1 + x^2 , if & 0 \leq x \leq 1 \\ 2 - x , if & x > 1\end{cases}\]
Discuss the continuity of the function f(x) at the point x = 0, where \[f\left( x \right) = \begin{cases}x, x > 0 \\ 1, x = 0 \\ - x, x < 0\end{cases}\]
For what value of k is the following function continuous at x = 1? \[f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k , & x = 1\end{cases}\]
If \[f\left( x \right) = \begin{cases}\frac{2^{x + 2} - 16}{4^x - 16}, \text{ if } & x \neq 2 \\ k , \text{ if } & x = 2\end{cases}\] is continuous at x = 2, find k.
Find the value of k for which \[f\left( x \right) = \begin{cases}\frac{1 - \cos 4x}{8 x^2}, \text{ when} & x \neq 0 \\ k ,\text{ when } & x = 0\end{cases}\] is continuous 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; \[f\left( x \right) = \begin{cases}k x^2 , & x \geq 1 \\ 4 , & x < 1\end{cases}\]at x = 1
Discuss the continuity of the f(x) at the indicated points:
(i) f(x) = | x | + | x − 1 | at x = 0, 1.
Discuss the continuity of the f(x) at the indicated points: f(x) = | x − 1 | + | x + 1 | at x = −1, 1.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 + x^3 + 2 x^2}{\tan^{- 1} x}, & \text{ if } x \neq 0 \\ 10 , & \text{ if } x = 0\end{cases}\]
Find all point of discontinuity of the function
The function \[f\left( x \right) = \begin{cases}\frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ 0 , & x = 0\end{cases}\]
The values of the constants a, b and c for which the function \[f\left( x \right) = \begin{cases}\left( 1 + ax \right)^{1/x} , & x < 0 \\ b , & x = 0 \\ \frac{\left( x + c \right)^{1/3} - 1}{\left( x + 1 \right)^{1/2} - 1}, & x > 0\end{cases}\] may be continuous at x = 0, are
Show that the function
(i) differentiable at x = 0, if m > 1
(ii) continuous but not differentiable at x = 0, if 0 < m < 1
(iii) neither continuous nor differentiable, if m ≤ 0
Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.
Is every continuous function differentiable?
Give an example of a function which is continuos but not differentiable at at a point.
The function f (x) = e−|x| is
If \[f\left( x \right) = \sqrt{1 - \sqrt{1 - x^2}},\text{ then } f \left( x \right)\text { is }\]
The function f (x) = |cos x| is
Let \[f\left( x \right) = \begin{cases}1 , & x \leq - 1 \\ \left| x \right|, & - 1 < x < 1 \\ 0 , & x \geq 1\end{cases}\] Then, f is
Find whether the following function is differentiable at x = 1 and x = 2 or not : \[f\left( x \right) = \begin{cases}x, & & x < 1 \\ 2 - x, & & 1 \leq x \leq 2 \\ - 2 + 3x - x^2 , & & x > 2\end{cases}\] .
Discuss the continuity of f at x = 1 ,
Where f(x) = `(3 - sqrt(2x + 7))/(x - 1)` for x = ≠ 1
= `(-1)/3` for x = 1
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
Find the points of discontinuity , if any for the function : f(x) = `(x^2 - 9)/(sinx - 9)`
If the function f (x) = `(15^x - 3^x - 5^x + 1)/(x tanx)`, x ≠ 0 is continuous at x = 0 , then find f(0).
Examine the continuity of the following function :
`{:(,f(x),=(x^2-16)/(x-4),",","for "x!=4),(,,=8,",","for "x=4):}} " at " x=4`
Discuss the continuity of the function f at x = 0, where
f(x) = `(5^x + 5^-x - 2)/(cos2x - cos6x),` for x ≠ 0
= `1/8(log 5)^2,` for x = 0
Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if" x ≠ 0),(0",", "if" x = 0):}` is discontinuous at x = 0.
The function f(x) = |x| + |x – 1| is ______.
The number of points at which the function f(x) = `1/(log|x|)` is discontinuous is ______.
f(x) = `{{:(3x + 5",", "if" x ≥ 2),(x^2",", "if" x < 2):}` at x = 2
f(x) = `{{:((1 - cos 2x)/x^2",", "if" x ≠ 0),(5",", "if" x = 0):}` at x = 0
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
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.
