If fx-x3-3 x2-4 x-b sin x-c cos x -b-c - is a one-one function - x - then the value of -b2-c2- is-where - represents greatest integer function

Here, f(x)=x^3+3 x^2+4 x+b sin x+c cos x ⇒ f^'(x)=3 x^2+6 x+4+b cos x c sin x Now nbsp;for f(x) to be one one function, the only possibility is f^'(x) gt; ...

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

Byju's Answer

Standard XII

Mathematics

One - One function

If fx=x3+3 x2...

Question

If $f (x) = x^{3} + 3 x^{2} + 4 x + b sin x + c cos x, b, c \in R$ is a one-one function $\forall x \in R$ , then the value of $[b^{2} + c^{2}]$ is
(where $[\cdot]$ represents greatest integer function)

Open in App

Solution

Here, $f (x) = x^{3} + 3 x^{2} + 4 x + b sin x + c cos x$
$\Rightarrow f^{'} (x) = 3 x^{2} + 6 x + 4 + b cos x - c sin x$

Now for $f (x)$ to be one-one function, the only possibility is
$f^{'} (x) > 0 \forall x \in R$
$\Rightarrow 3 x^{2} + 6 x + 4 + b cos x - c sin x > 0 \forall x \in R$
$\Rightarrow 3 x^{2} + 6 x + 4 > c sin x - b cos x, \forall x \in R$
$\Rightarrow 3 x^{2} + 6 x + 4 > \sqrt{b^{2} + c^{2}}, \forall x \in R$
$\Rightarrow \sqrt{b^{2} + c^{2}} < 3 (x^{2} + 2 x + 1) + 1, \forall x \in R$
$\Rightarrow \sqrt{b^{2} + c^{2}} < 3 (x + 1)^{2} + 1, \forall x \in R$
$\Rightarrow \sqrt{b^{2} + c^{2}} < 1, \forall x \in R$
$\Rightarrow b^{2} + c^{2} < 1, \forall x \in R$
Also, $b^{2} + c^{2} \geq 0$
$\Rightarrow 0 \leq b^{2} + c^{2} < 1, \forall x \in R$
$∴ [b^{2} + c^{2}] = 0$

Suggest Corrections

Similar questions

Q. Let

A = {x : [5 sin x] + [cos x] + 6 = 0, x \in R}

, where

[.]

represents the greatest integer function. If

f (x) = \sqrt{3} sin x + cos x \forall x \in A

, then

f (x) = x^{3} + 3 x^{2} + 4 x + b sin x + c cos x, \forall x \in R

is a one-one function, then the value of

b^{2} + c^{2}

Q. If

f (x) = x^{3} + 3 x^{2} + 4 x + b sin x + c cos x \forall x \in R

is a one-one function, then the maximum value of

(b^{2} + c^{2})

Q. Let

f : R \to R

be defined as

f (x) = (| x - 1 | + | 4 x - 11 |) [x^{2} - 2 x - 2],

where

[.]

denotes the greatest integer function. Then the number of points of discontinuity of

f (x)

(\frac{1}{2}, \frac{5}{2})

Q. Let the function

f (x)

be defined as follows

f (x) = {\begin{matrix} | x | \forall x \in [- 2, 2] [x] \forall x \in [2, 4] \end{matrix}

, where

[\cdot]

represents the greatest integer function. Then

Join BYJU'S Learning Program

If f(x)=x3+3x2+4x+bsinx+ccosx,b,c∈R is a one-one function ∀x∈R, then the value of [b2+c2] is​​​​​​​ (where [⋅] represents greatest integer function)

If $f (x) = x^{3} + 3 x^{2} + 4 x + b sin x + c cos x, b, c \in R$ is a one-one function $\forall x \in R$ , then the value of $[b^{2} + c^{2}]$ is
(where $[\cdot]$ represents greatest integer function)