Let \(f\) be a differentiable function such that \(f(x+y) = f(x) + f(y)\) for all \(x, y \in R\). Which of the following statements is/are correct? I. If \(f(x) = x\,g(x)\), then the derivative of \(f(x)\) with respect to \(x\) is equal to \(g(0)\). II. If \(f(x) = x^2 g(x)\), then the derivative of \(f(x)\) with respect to \(x\) is equal to \(0\). (Here \(g\) is a continuous function) Select the answer using the code given below.
Both I and II
Since \(f\) is differentiable and satisfies \(f(x+y)=f(x)+f(y)\), it must be of the form \(f(x)=cx\) for some constant \(c\). If \(f(x)=xg(x)\) with \(g\) continuous, then \(g(x)=c\) for all \(x \neq 0\), and by continuity \(g(0)=c=f'(x)\), so statement I holds. If \(f(x)=x^2g(x)\) with \(g\) continuous, then \(g(x)=c/x\) for \(x \neq 0\), which can only be continuous at \(0\) if \(c=0\), forcing \(f \equiv 0\) and hence \(f'(x)=0\), so statement II also holds. Thus both statements are correct.
Similar Questions
The set of all points, where the function \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {1 - {{\rm{e}}^{ - {{\rm{x}}^2}}}} \) is differentiable, is
Consider the function
\( f(x)=\begin{cases} x^2\ln|x|, & x\neq 0,\\[4pt] 0, & x=0. \end{cases} \)
What is \(f'(0)\) equal to?
The left-hand derivative of f(x) = [x] sin (πx) at x = k
Where k is an integer and [x] is the greatest integer function, isIf \({\rm{f}}\left( {\rm{x}} \right) = {\rm{x}}\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} + 1} } \right)\) , then f(x) is
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {3{{\rm{x}}^2} + 12{\rm{x}} - 1,{\rm{\;\;}} - 1 \le {\rm{x}} \le 2}\\ {37 - {\rm{x}},{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}2 < {\rm{x}} \le 3} \end{array}} \right.\)
Which of the following statements is/are correct?
1. f(x) is increasing in the interval [-1, 2]
2. f(x) is decreasing in the interval (2, 3).
Select the correct answer using the code given below:
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {3{{\rm{x}}^2} + 12{\rm{x}} - 1,{\rm{\;\;}} - 1 \le {\rm{x}} \le 2}\\ {37 - {\rm{x}},{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}2 < {\rm{x}} \le 3} \end{array}} \right.\)
Which of the following statements are correct?
1. f(x) is continuous at x = 2
2. f(x) attains greatest value at x = 2
3. f(x) is differentiable at x = 2
Select the correct answer using the code given below:
Consider the following statements:
1. The function f(x) is continuous at x = 0
2. The function f(x) is continuous at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\)
Which of the above statements is/are correct?Consider the following statements:
1. The function f(x) is differentiable at x = 0
2. The function f(x) is differentiable at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\) .
Which of the above statements is/are correct?What is f’(4) equal to?
What is f’’(2.5) equal to?
Important Questions from Differentiability
What is the value of f'(x) at x = 4 from the following table of values?
x 1 2 3 4 f(x) 20 22 27 35 The set of all points, where the function \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {1 - {{\rm{e}}^{ - {{\rm{x}}^2}}}} \) is differentiable, is
Let f be a differentiable function defined for all x ∈ R such that f(x3) = x5 for all x ∈ R, x ≠ 0. Then the value of \(\dfrac{df}{dx} (8)\) is:
If \(f(x)=\displaystyle\sum_{n-0}^{2k}\left(a_n|x|^n+b_n\ \sin^2x\right)\), where \(a_i^{'}\)s and \(b_i^{'}\)s (0 ≤ i ≤ k) are real constants, then f(x) is:
The set of all point where the function f(x) = 2x|x| is differentiable, is:
