Advertisements
Advertisements
Question
The function f(x) = `"e"^|x|` is ______.
Options
Continuous everywhere but not differentiable at x = 0
Continuous and differentiable everywhere
Not continuous at x = 0
None of these
Advertisements
Solution
The function f(x) = `"e"^|x|` is continuous everywhere but not differentiable at x = 0.
Explanation:
Given that: f(x) = `"e"^|x|`
We know that modulus function is continuous but not differentiable in its domain.
Let g(x) = |x| and t(x) = ex
∴ f(x) = got(x) = g[t(x)] = `"e"^|x|`
Since g(x) and t(x) both are continuous at x = 0 but f(x) is not differentiable at x = 0.
APPEARS IN
RELATED QUESTIONS
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx +1", if" x<= pi),(cos x", if" x > pi):}` at x = π
Show that the function defined by f(x) = cos (x2) is a continuous function.
Show that the function defined by f(x) = |cos x| is a continuous function.
Determine the value of the constant k so that the function
\[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0 .\]
If \[f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, find k.
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
Discuss the continuity of the function
Find the points of discontinuity, if any, of the following functions:
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}2 , & \text{ if } x \leq 3 \\ ax + b, & \text{ if } 3 < x < 5 \\ 9 , & \text{ if } x \geq 5\end{cases}\]
The function \[f\left( x \right) = \begin{cases}x^2 /a , & \text{ if } 0 \leq x < 1 \\ a , & \text{ if } 1 \leq x < \sqrt{2} \\ \frac{2 b^2 - 4b}{x^2}, & \text{ if } \sqrt{2} \leq x < \infty\end{cases}\] is continuous on (0, ∞), then find the most suitable values of a and b.
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.
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
\[f\left( x \right) = \frac{\left( 27 - 2x \right)^{1/3} - 3}{9 - 3 \left( 243 + 5x \right)^{1/5}}\left( x \neq 0 \right)\] is continuous, is given by
The function
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
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
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 \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
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
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
Let f(x) = |sin x|. Then ______.
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
A real value of x satisfies `((3 - 4ix)/(3 + 4ix))` = α – iβ (α, β ∈ R), if α2 + β2 is equal to
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
The function f(x) = x2 – sin x + 5 is continuous at x =
Discuss the continuity of the following function:
f(x) = sin x – cos x
If \[f\] and \[g\] are continuous at \[x=c\], which collection is continuous at \[x=c\]?
If \[g\] is continuous and \[g(x)\neq 0\], which function is continuous?
Which conditions guarantee that \[f\circ g\] is continuous at \[c\]?
