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Differentiability

If fx=x3+3x2+...

Question

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

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Solution

$f (x) = x^{3} + 3 x^{2} + 4 x + b sin x + c cos x$
Differentiating w.r.t. $x,$ we get
$f^{'} (x) = 3 x^{2} + 6 x + 4 + b cos x - c sin x$
For $f (x)$ to be one-one, only possibility is $f^{'} (x) \geq 0 \forall x \in R$

$3 x^{2} + 6 x + 4 + b cos x - c sin x \geq 0$ for all $x \in R$
$\Rightarrow 3 x^{2} + 6 x + 4 \geq c sin x - b cos x$ for all $x \in R$
$\Rightarrow 3 x^{2} + 6 x + 4 \geq \sqrt{b^{2} + c^{2}}$
$\Rightarrow \sqrt{b^{2} + c^{2}} \leq 3 (x^{2} + 2 x + 1) + 1$
$\Rightarrow \sqrt{b^{2} + c^{2}} \leq 3 (x + 1)^{2} + 1$
$\Rightarrow \sqrt{b^{2} + c^{2}} \leq 1$
$\Rightarrow b^{2} + c^{2} \leq 1$

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