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Quantitative Aptitude

Functions

19. STATEMENT...

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19. STATEMENT 1 : Integral of an even function is not always an odd function. STATEMENT 2: Integral of an odd function is an even function. Are Statement 1 and 2 true?

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Which of the following statements is/are true?

I

: Every function must be either even or odd function

I I

f (x) = log (x + \sqrt{x^{2} + 1})

is an odd function.

f (x)

is an odd function then-
(i)

\frac{f (- x) + f (x)}{2}

is an even function
(ii)

[∣ f (x) ∣ + 1]

is even where [.] denotes greatest integer function.
(iii)

\frac{f (x) - f (- x)}{2}

is neither even nor odd
(iv)

f (x) + f (- x)

is neither even nor odd
Which of these statements are correct

Q. Assertion :The function

f (x) = \int_{0}^{x} \sqrt{1 + t^{2}} d t

is an odd function and

g (x) = f^{'} (x)

is an even function. Reason: For a differentiable function

f (x)

f^{''} (x)

is an even function, then

f (x)

is an odd function.

Q. Prove that the derivative of an odd function is always an even function.

f (x) = \frac{cos x}{[\frac{x}{π}] + \frac{1}{2}},

x

is not an integral multiple of

π

[.]

denotes the greatest integer function, then

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