Advertisements
Advertisements
Question
The function f(x) = x |x| is ______.
Options
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
Solution
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.
RELATED QUESTIONS
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.
Find the value of k if f(x) is continuous at x = π/2, where \[f\left( x \right) = \begin{cases}\frac{k \cos x}{\pi - 2x}, & x \neq \pi/2 \\ 3 , & x = \pi/2\end{cases}\]
Discuss the continuity of the function
If \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}\]
for x ≠ π/4, find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].
Show that the function g (x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.
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.
Determine the value of the constant 'k' so that function f
\[f\left( x \right) = \begin{cases}\frac{\left| x^2 - x \right|}{x^2 - x}, & x \neq 0, 1 \\ 1 , & x = 0 \\ - 1 , & x = 1\end{cases}\] then f (x) is continuous for all
If \[f\left( x \right) = \begin{cases}\frac{1 - \sin x}{\left( \pi - 2x \right)^2} . \frac{\log \sin x}{\log\left( 1 + \pi^2 - 4\pi x + 4 x^2 \right)}, & x \neq \frac{\pi}{2} \\ k , & x = \frac{\pi}{2}\end{cases}\]is continuous at x = π/2, then k =
If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to
If the function \[f\left( x \right) = \frac{2x - \sin^{- 1} x}{2x + \tan^{- 1} x}\] is continuous at each point of its domain, then the value of f (0) is
If \[f\left( x \right) = x \sin\frac{1}{x}, x \neq 0,\]then the value of the function at x = 0, so that the function is continuous at x = 0, is
The value of a for which the function \[f\left( x \right) = \begin{cases}5x - 4 , & \text{ if } 0 < x \leq 1 \\ 4 x^2 + 3ax, & \text{ if } 1 < x < 2\end{cases}\] is continuous at every point of its domain, is
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 \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
Let f (x) = |cos x|. Then,
The function f (x) = 1 + |cos x| is
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
The function f(x) = `"e"^|x|` is ______.
Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.
If `f(x) = {{:(-x^2",", "when" x ≤ 0),(5x - 4",", "when" 0 < x ≤ 1),(4x^2 - 3x",", "when" 1 < x < 2),(3x + 4",", "when" x ≥ 2):}`, then
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx^2",", if x ≤ 2),(3",", if x > 2):}` at x = 2
Find the values of `a` and ` b` such that the function by:
`f(x) = {{:(5",", if x ≤ 2),(ax + b",", if 2 < x < 10),(21",", if x ≥ 10):}`
is a continuous function.
If \[g\] is continuous and \[g(x)\neq 0\], which function is continuous?
Why is \[f(x)=\sin(x^2)\] continuous for all real \[x\]?
Why is \[f(x)=|1-x+|x||\] continuous for every real \[x\]?
