If fx - x3- 4x - 1 - x - 0 and gx is image of fx with respect to line y - x then

The correct options are A g(1) = 2 B g'(1)=1/8 C g”(1)= 3/16 ∴ g(x)=f^ 1(x) ⇒ f(g(x))=x⇒ g'(x)=1/f'(g(x)) g'(1)=1/f'(g(1))=1/f'( ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Inverse of a Function

If fx = x3-...

Question

If $f (x) = x^{3} - 4 x + 1 \forall x > 0$ and $g (x)$ is image of $f (x)$ with respect to line $y = x$ then

g (1) = 2

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

g^{'} (1) = \frac{1}{8}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

f^{''} (g (1)) = 12

No worries! We‘ve got your back. Try BYJU‘S free classes today!

g^{''} (1) = - \frac{3}{16}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct options are
A $g (1) = 2$
B $g^{'} (1) = \frac{1}{8}$
C $g^{''} (1) = - \frac{3}{16}$
$∴ g (x) = f^{- 1} (x)$
$\Rightarrow f (g (x)) = x \Rightarrow g^{'} (x) = \frac{1}{f^{'} (g (x))}$
$g^{'} (1) = \frac{1}{f^{'} (g (1))} = \frac{1}{f^{'} (2)} = \frac{1}{8}$
$g " (x) = - {\frac{f " g (x) . g^{'} (x)}{f^{'} g (x)}}^{2}$
$g " (1) = - \frac{f " (2) . g^{'} (1)}{f^{'} g (1)} = \frac{- 12 (\frac{1}{8})}{8} = \frac{- 3}{16}$

Suggest Corrections

Similar questions

Q. If

f (x)

and

g (x)

are twice differentiable functions on

(0, 3)

satisfying

f^{''} (x) = g^{''}, f^{'} (1) = 4, g^{'} (1) = 6, f (2) = 3, g (2) = 9

, then

f (1) - g (1)

is:

If $f (x) = l o g_{e} (1 - x)$ and $g (x) = [x$ , then determine each of the following functions :

(i) $f + g$
(ii) $f g$
(iii) $\frac{f}{g}$
(iv) $\frac{g}{f}$
Also, find $(f + g) (- 1), (f g) (0), (\frac{f}{g}) (\frac{1}{2}), (\frac{g}{f}) (\frac{1}{2})$

g (x) = {lim}_{m \to \infty} \frac{x^{m} f (x) + h (x) + 3}{2 x^{m} + 4 x + 1}

when

x \neq 1

and

g (1) = e^{3}

such that

f (x), g (x)

and

h (x)

are continuous function at

x = 1

and

f (1) - h (1) = a (b - g (1))

then

a + b

Q. Find f + g, f − g, cf (c ∈ R, c ≠ 0), fg,

\frac{1}{f} and \frac{f}{g}

in each of the following:

(a) If f(x) = x³ + 1 and g(x) = x + 1
(b) If

f (x) = \sqrt{x - 1}

and

g (x) = \sqrt{x + 1}

Find f + g, f - g, $c f (c ϵ R, c \neq 0)$ . fg. $\frac{1}{f}$ and $\frac{f}{g}$ in each of the following :
(i) $f (x) = x^{3} + 1$ and $g (x) = x + 1$
(ii) $f (x) = \sqrt{x - 1}$ and $g (x) = \sqrt{x + 1}$

Join BYJU'S Learning Program

If f(x)=x3−4x+1∀x>0 and g(x) is image of f(x) with respect to line y=x then

The correct options are A g(1)=2 B g′(1)=18 C g′′(1)=−316∴g(x)=f−1(x)⇒f(g(x))=x⇒g′(x)=1f′(g(x))g′(1)=1f′(g(1))=1f′(2)=18g"(x)=−f"g(x).g′(x)f′g(x)2g"(1)=−f"(2).g′(1)f′g(1)=−12(18)8=−316

If $f (x) = x^{3} - 4 x + 1 \forall x > 0$ and $g (x)$ is image of $f (x)$ with respect to line $y = x$ then