SATHEE: Application of Derivatives 2 Question 8

Application of Derivatives 2 Question 8

####8. If $f (x) = x^{3} + b x^{2} + c x + d$ and $0 < b^{2} < c$ , then in $(- \infty, \infty)$

(a) $f (x)$ is strictly increasing function

(2004, 2M)

(b) $f (x)$ has a local maxima

(d) $f (x)$ is bounded

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Answer:

(a)

Solution:

Given,

$f (x) = x^{3} + b x^{2} + c x + d$

$\Rightarrow f^{'} (x) = 3 x^{2} + 2 b x + c$

As we know that, if $a x^{2} + b x + c > 0, \forall x$ , then $a > 0$ and $D < 0$ .

Now, $D = 4 b^{2} - 12 c = 4 (b^{2} - c) - 8 c$

[where, $b^{2} - c < 0$ and $c > 0$ ]

$∴ D = (- ve) or D < 0$

$\Rightarrow f^{'} (x) = 3 x^{2} + 2 b x + c > 0 \forall x \in (- \infty, \infty)$ [as $D < 0$ and $a > 0$ ]

Hence, $f (x)$ is strictly increasing function.

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Application of Derivatives 2 Question 8

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