Advertisements
Advertisements
प्रश्न
The function f(x) = x |x| is ______.
पर्याय
continuous and differentiable at x = 0
continuous but not differentiable at x = 0
differentiable but not continuous at x = 0
neither differentiable nor continuous at x = 0
Advertisements
उत्तर
The function f(x) = x |x| is continuous and differentiable at x = 0.
Explanation:
Given, the function is f(x) = x |x| for x ∈ R.
The function can be written as,
f(x) = `{{:( x^2; x > 0),(-x^2; x ≤ 0):}`
Now, Rf(0) = `lim_(x rightarrow 0^+) (x^2)` = 0
and Lf(0) = `lim_(x rightarrow 0^-) (-x^2)` = 0
So, Lf(0) = Rf(0) = f(0)
So, the function is continuous at 0.
Now, Rf'(0) = `lim_(x rightarrow 0^+) (f(x) - f(0))/(x - 0)`
= `lim_(x rightarrow 0^+) (x^2 - 0)/x` = 0
and Lf'(0) = `lim_(x rightarrow 0^-) (f(x) - f(0))/(x - 0)`
= `lim_(x rightarrow 0^-) (-x^2 - 0)/x` = 0
So, Lf'(0) = Rf'(0)
So, function is differentiable at 0.
संबंधित प्रश्न
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 following function:
f(x) = sin x × cos x
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.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2kx}{x^2}, \text{ if } & x \neq 0 \\ 8 , \text{ if } & x = 0\end{cases}\] at x = 0
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: \[f\left( x \right) = \begin{cases}kx + 5, & \text{ if } x \leq 2 \\ x - 1, & \text{ if } x > 2\end{cases}\]
Discuss the continuity of f(x) = sin | x |.
Show that f (x) = | cos x | is a continuous function.
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).
\[f\left( x \right) = \frac{\left( 27 - 2x \right)^{1/3} - 3}{9 - 3 \left( 243 + 5x \right)^{1/5}}\left( x \neq 0 \right)\] is continuous, is given by
The value of a for which the function \[f\left( x \right) = \begin{cases}\frac{\left( 4^x - 1 \right)^3}{\sin\left( x/a \right) \log \left\{ \left( 1 + x^2 /3 \right) \right\}}, & x \neq 0 \\ 12 \left( \log 4 \right)^3 , & x = 0\end{cases}\]may be continuous at x = 0 is
The function
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 =
Find the values of a and b so that the function
Find the values of a and b, if the function f defined by
If f is defined by \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]
If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of
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
`lim_("x" -> 0) ("x cos x" - "log" (1 + "x"))/"x"^2` is equal to ____________.
`lim_("x" -> 0) (1 - "cos" 4 "x")/"x"^2` is equal to ____________.
`lim_("x" -> 0) (1 - "cos x")/"x sin x"` is equal to ____________.
The value of f(0) for the function `f(x) = 1/x[log(1 + x) - log(1 - x)]` to be continuous at x = 0 should be
What is the values of' 'k' so that the function 'f' is continuous at the indicated point
Discuss the continuity of the following function:
f(x) = sin x – cos x
