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Standard XII

Mathematics

nth Derivative Test

lf f : R→...

Question

lf $f$ : $R \to R$ is an even function having derivatives of all orders then an odd function among the following is

f^{^{''}}

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f^{^{'''}}

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f^{^{'}} + f^{^{''}}

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f^{^{'}} + f^{^{'''}}

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Solution

The correct option is B $f^{^{'''}}$
Fact: If f is an even function and having derivatives of all orders then,
$f^{'}, f^{'''}, f^{'''''}, . . . . . . . . . . . . . . . . . . f^{n}$ where $n$ is odd are odd function
and $f, f^{''}, f^{''''}, . . . . . . . . . . . . . f^{m}$ where $m$ is even are even function
And vice-versa if f is odd function.

Suggest Corrections

Similar questions

Left hand derivative and right hand derivative of a function

f (x)

at a point

x = a

are defined as

f^{'} (a^{-}) = lim h \to 0^{+} \frac{f (a) - f (a - h)}{h} = lim h \to 0^{-} \frac{f (a) - f (a - h)}{h} = lim x \to a^{+} \frac{f (a) - f (x)}{a - x}

respectively

Let

f

be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.

f

is odd, which of the following is Left-hand derivative of

f

x = a

Left hand derivative and right hand derivative of a function

f (x)

at a point

x = a

are defined as

f^{'} (a^{-}) = lim h \to 0^{+} \frac{f (a) - f (a - h)}{h} = lim h \to 0^{-} \frac{f (a) - f (a - h)}{h} = lim x \to a^{+} \frac{f (a) - f (x)}{a - x}

respectively

Let

f

be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.

f

is even, which of the following is Right hand derivative of

f^{'}

x = a

Left hand derivative and right hand derivative of a function

f (x)

at a point

x = a

are defined as

f^{'} (a^{-}) = lim h \to 0^{+} \frac{f (a) - f (a - h)}{h} = lim h \to 0^{-} \frac{f (a) - f (a - h)}{h} = lim x \to a^{+} \frac{f (a) - f (x)}{a - x}

respectively

Let

f

be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function.

The statement

lim h \to 0 \frac{f (- x) - f (- x - h)}{h} = lim h \to 0 \frac{f (x) - f (x - h)}{- h}

implies that for all x

ϵ R

Q. If

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.

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lf f : R→R is an even function having derivatives of all orders then an odd function among the following is

lf $f$ : $R \to R$ is an even function having derivatives of all orders then an odd function among the following is