Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given:
It is given that the given function is differentiable at x = 1.
We know every differentiable function is continuous. Therefore it is continuous at x =1. Then,
\[\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) \]
\[ \Rightarrow \lim_{x \to 1} a x^2 - b = \lim_{x \to 1} \frac{1}{x}\]
\[ \Rightarrow a - b = 1 . . . (i)\]
It is also differentiable at x=1. Therefore,
(LHD at x = 1) = (RHD at x = 1)
\[\Rightarrow \lim_{x \to 1^-} \frac{f(x) - f(1)}{x - 1} = \lim_{x \to 1^+} \frac{f(x) - f(1)}{x - 1}\]
\[ \Rightarrow \lim_{x \to 1} \frac{a x^2 - b - 1}{x - 1} = \lim_{x \to 1} \frac{\frac{1}{x} - 1}{x - 1} \]
\[ \Rightarrow \lim_{x \to 1} \frac{a x^2 + 1 - a - 1}{x - 1} = \lim_{x \to 1} \frac{- (x - 1)}{x - 1} \left[ \text { Using }(i) \right]\]
\[ \Rightarrow \lim_{x \to 1} a (x + 1) = \lim_{x \to 1} - 1 \]
\[ \Rightarrow 2a = - 1 \]
\[ \Rightarrow a = - \frac{1}{2}\]
From (i), we have:
\[a - b = 1\]
\[ \Rightarrow - \frac{1}{2} - b = 1\]
\[ \Rightarrow b = - \frac{3}{2}\]
Hence, when
the function is differentiable at x = 1.
APPEARS IN
संबंधित प्रश्न
Find the value of 'k' if the function
`f(X)=(tan7x)/(2x) , "for " x != 0 `
`=k`, for x=0
is continuos at x=0
Examine the following function for continuity:
f(x) = `1/(x - 5)`, x ≠ 5
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):}`
Discuss the continuity of the following functions at the indicated point(s):
Show that
\[f\left( x \right) = \begin{cases}1 + x^2 , if & 0 \leq x \leq 1 \\ 2 - x , if & x > 1\end{cases}\]
Show that
\[f\left( x \right) = \begin{cases}\frac{\sin 3x}{\tan 2x} , if x < 0 \\ \frac{3}{2} , if x = 0 \\ \frac{\log(1 + 3x)}{e^{2x} - 1} , if x > 0\end{cases}\text{is continuous at} x = 0\]
Find the value of 'a' for which the function f defined by
Discuss the continuity of the function f(x) at the point x = 1/2, where \[f\left( x \right) = \begin{cases}x, 0 \leq x < \frac{1}{2} \\ \frac{1}{2}, x = \frac{1}{2} \\ 1 - x, \frac{1}{2} < x \leq 1\end{cases}\]
For what value of k is the function
\[f\left( x \right) = \begin{cases}\frac{\sin 5x}{3x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0?\]
Determine the values of a, b, c for which the function f(x) = `{((sin(a + 1)x + sin x)/x, "for" x < 0),(x, "for" x = 0),((sqrt(x + bx^2) - sqrtx)/(bx^(3"/"2)), "for" x > 0):}` is continuous at x = 0.
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos kx}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\text{is continuous at x} = 0, \text{ find } k .\]
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: \[f\left( x \right) = \begin{cases}\frac{x^4 + x^3 + 2 x^2}{\tan^{- 1} x}, & \text{ if } x \neq 0 \\ 10 , & \text{ if } x = 0\end{cases}\]
Find all the points of discontinuity of f defined by f (x) = | x |− | x + 1 |.
Let f (x) = | x | + | x − 1|, then
Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:
Show that the function
\[f\left( x \right) = \begin{cases}\left| 2x - 3 \right| \left[ x \right], & x \geq 1 \\ \sin \left( \frac{\pi x}{2} \right), & x < 1\end{cases}\] is continuous but not differentiable at x = 1.
If f is defined by f (x) = x2, find f'(2).
Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]
The set of points where the function f (x) = x |x| is differentiable is
The function f (x) = e−|x| is
Examine the continuity of f(x)=`x^2-x+9 "for" x<=3`
=`4x+3 "for" x>3, "at" x=3`
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) = x2 - x + 9, for x ≤ 3
= 4x + 3, for x > 3
at x = 3.
If y = ( sin x )x , Find `dy/dx`
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
Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`
Discuss the continuity of the function f(x) = sin x . cos x.
The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.
The function f(x) = |x| + |x – 1| is ______.
The number of points at which the function f(x) = `1/(log|x|)` is discontinuous is ______.
A continuous function can have some points where limit does not exist.
f(x) = `{{:(3x + 5",", "if" x ≥ 2),(x^2",", "if" x < 2):}` at x = 2
f(x) = `{{:((sqrt(1 + "k"x) - sqrt(1 - "k"x))/x",", "if" -1 ≤ x < 0),((2x + 1)/(x - 1)",", "if" 0 ≤ x ≤ 1):}` at x = 0
Examine the differentiability of f, where f is defined by
f(x) = `{{:(1 + x",", "if" x ≤ 2),(5 - x",", "if" x > 2):}` at x = 2
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).
