Advertisements
Advertisements
प्रश्न
Is every continuous function differentiable?
उत्तर
No, function may be continuous at a point but may not be differentiable at that point .
For example: function
APPEARS IN
संबंधित प्रश्न
Examine the following function for continuity:
f (x) = x – 5
Examine the following function for continuity:
`f(x) = (x^2 - 25)/(x + 5), x != -5`
Discuss the continuity of the function f, where f is defined by `f(x) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`
Discuss the continuity of the function f, where f is defined by `f(x) = {(2x , ","if x < 0),(0, "," if 0 <= x <= 1),(4x, "," if x > 1):}`
If \[f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}; for & x \neq 1 \\ 2 ; for & x = 1\end{cases}\] Find whether f(x) is continuous at x = 1.
Discuss the continuity of the following functions at the indicated point(s): (iv) \[f\left( x \right) = \left\{ \begin{array}{l}\frac{e^x - 1}{\log(1 + 2x)}, if & x \neq a \\ 7 , if & x = 0\end{array}at x = 0 \right.\]
Show that
\[f\left( x \right) = \begin{cases}\frac{\sin 3x}{\tan 2x} , if x < 0 \\ \frac{3}{2} , if x = 0 \\ \frac{\log(1 + 3x)}{e^{2x} - 1} , if x > 0\end{cases}\text{is continuous at} x = 0\]
Determine the value of the constant k so that the function
\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.\]
If \[f\left( x \right) = \begin{cases}\frac{x - 4}{\left| x - 4 \right|} + a, \text{ if } & x < 4 \\ a + b , \text{ if } & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, \text{ if } & x > 4\end{cases}\] is continuous at x = 4, find a, b.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{x^2 - 25}{x - 5}, & x \neq 5 \\ k , & x = 5\end{cases}\]at x = 5
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\left| x - 3 \right|, & \text{ if } x \geq 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & \text{ if } x < 1\end{cases}\]
Find f (0), so that \[f\left( x \right) = \frac{x}{1 - \sqrt{1 - x}}\] becomes continuous at x = 0.
Write the value of b for which \[f\left( x \right) = \begin{cases}5x - 4 & 0 < x \leq 1 \\ 4 x^2 + 3bx & 1 < x < 2\end{cases}\] is continuous at x = 1.
The function
If \[f\left( x \right) = \begin{cases}\frac{{36}^x - 9^x - 4^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ k , & x = 0\end{cases}\]is continuous at x = 0, then k equals
If \[f\left( x \right) = \begin{cases}\frac{\sin \left( \cos x \right) - \cos x}{\left( \pi - 2x \right)^2}, & x \neq \frac{\pi}{2} \\ k , & x = \frac{\pi}{2}\end{cases}\]is continuous at x = π/2, then k is equal to
If \[f\left( x \right) = \begin{cases}a x^2 - b, & \text { if }\left| x \right| < 1 \\ \frac{1}{\left| x \right|} , & \text { if }\left| x \right| \geq 1\end{cases}\] is differentiable at x = 1, find a, b.
Define differentiability of a function at a point.
The function f (x) = e−|x| is
If \[f\left( x \right) = \sqrt{1 - \sqrt{1 - x^2}},\text{ then } f \left( x \right)\text { is }\]
If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]
Find whether the following function is differentiable at x = 1 and x = 2 or not : \[f\left( x \right) = \begin{cases}x, & & x < 1 \\ 2 - x, & & 1 \leq x \leq 2 \\ - 2 + 3x - x^2 , & & x > 2\end{cases}\] .
Examine the continuity of f(x)=`x^2-x+9 "for" x<=3`
=`4x+3 "for" x>3, "at" x=3`
Find the value of k for which the function f (x ) = \[\binom{\frac{x^2 + 3x - 10}{x - 2}, x \neq 2}{ k , x^2 }\] is continuous at x = 2 .
The total cost C for producing x units is Rs (x2 + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.
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.
Discuss the continuity of function f at x = 0.
Where f(X) = `[ [sqrt ( 4 + x ) - 2 ]/ ( 3x )]`, For x ≠ 0
= `1/12`, For x = 0
The probability distribution function of continuous random variable X is given by
f( x ) = `x/4`, 0 < x < 2
= 0, Otherwise
Find P( x ≤ 1)
If the function f is continuous at x = 2, then find 'k' where
f(x) = `(x^2 + 5)/(x - 1),` for 1< x ≤ 2
= kx + 1 , for x > 2
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
y = |x – 1| is a continuous function.
f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
f(x) = `{{:(x^2/2",", "if" 0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",", "if" 1 < x ≤ 2):}` at x = 1
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).
If f is continuous on its domain D, then |f| is also continuous on D.
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:
