Let be a real function. Number of correct statements among the following: Statement I: If is an odd function on and is known to be an increasing function for then it is also decreasing function for . Statement II: If is an odd function and exists

Question

Let $f (x)$ be a real function.

Number of correct statements among the following:

Statement I: If $f (x)$ is an odd function on $R$ and is known to be an increasing function for $x > 0$ then it is also decreasing function for $x < 0$ .

Statement II: If $f (x)$ is an odd function and $\lim_{x \to 0} f (x)$ exists then $\lim_{x \to 0} f (x) = 0$ .

Statement III: If $f (x)$ is an even function on $R$ and is known to be increasing for $x > 0$ , then it is decreasing function for $x < 0$ .

Statement IV: If $f (x)$ is an even function and $f (0)$ exists then $\lim_{x \to 0^{+}} f (x) = \lim_{x \to 0^{-}} f (x)$ .

Smart Achievers · Accepted Answer

(2)

$Statement-I: f (x) = x^{3} Not satisfies$

$Statement-II: For odd if \lim_{x \to 0} f (x) exists it is 0$

$Statement-III: from graph, it is correct$

$\lim_{x \to 0} f (x)$ ${\begin{matrix} \cos (\frac{1}{x}) & x \neq 0 \\ 0 & x = 0 \end{matrix}$ $Not satisfies$

Solution

1 1

2 2

3 0

4 3

Ans.
(2)

$Statement-I: f (x) = x^{3} Not satisfies$

$Statement-II: For odd if \lim_{x \to 0} f (x) exists it is 0$

$Statement-III: from graph, it is correct$

$Statement-IV: f (x) =$ ${\begin{matrix} \cos (\frac{1}{x}) & x \neq 0 \\ 0 & x = 0 \end{matrix}$ $Not satisfies$

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Solution

1 1

2 2

3 0

4 3

Ans. (2) Statement-I: f(x)=x3 Not satisfies Statement-II: For odd if limx→0f(x) exists it is 0 Statement-III: from graph, it is correct Statement-IV: f(x)={cos(1x)x≠00x=0 Not satisfies

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Ans.
(2)

$Statement-I: f (x) = x^{3} Not satisfies$

$Statement-II: For odd if \lim_{x \to 0} f (x) exists it is 0$

$Statement-III: from graph, it is correct$

$Statement-IV: f (x) =$ ${\begin{matrix} \cos (\frac{1}{x}) & x \neq 0 \\ 0 & x = 0 \end{matrix}$ $Not satisfies$