SATHEE: Application of Derivatives 4 Question 33

Application of Derivatives 4 Question 33

35. If $f (x)$ is a cubic polynomial which has local maximum at $x = - 1$ . If $f (2) = 18, f (1) = - 1$ and $f^{'} (x)$ has local minimum at $x = 0$ , then

(2006, 3M)

(a) the distance between $(- 1, 2)$ and $(a, f (a))$ , where $x = a$ is the point of local minima, is $2 \sqrt{5}$

(b) $f (x)$ is increasing for $x \in [1, 2 \sqrt{5}]$

(d) the value of $f (0) = 5$

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Answer:

Correct Answer: 35. (a)

Solution:

Since, $f (x)$ has local maxima at $x = - 1$ and $f^{'} (x)$ has local minima at $x = 0$ .

$∴ f^{''} (x) = λ x$

On integrating, we get

$\begin{aligned} f^{'} (x) = λ \frac{x^{2}}{2} + c [∵ f^{'} (- 1) = 0] \\ \Rightarrow & \frac{λ}{2} + c = 0 \Rightarrow λ = - 2 c \end{aligned}$

Again, integrating on both sides, we get

$f (x) = λ \frac{x^{3}}{6} + c x + d$

$\begin{array}{ll} \Rightarrow & f (2) = λ \frac{8}{6} + 2 c + d = 18 \\ and & f (1) = \frac{λ}{6} + c + d = - 1 \end{array}$

From Eqs. (i), (ii) and (iii),

$\begin{aligned} f (x) & = \frac{1}{4} (19 x^{3} - 57 x + 34) \\ ∴ f^{'} (x) = \frac{1}{4} (57 x^{2} - 57) & = \frac{57}{4} (x - 1) (x + 1) \end{aligned}$

For maxima or minima, put $f^{'} (x) = 0 \Rightarrow x = 1, - 1$

Now

$f^{''} (x) = \frac{1}{4} (114 x)$

$\begin{array}{ll} At & x = 1, f^{''} (x) > 0, minima \\ At & x = - 1, f^{''} (x) < 0, maxima \end{array}$

$∴ f (x)$ is increasing for $[1, 2 \sqrt{5}]$ .

$∴ f (x)$ has local maxima at $x = - 1$ and $f (x)$ has local minima at $x = 1$ .

Also, $f (0) = 34 / 4$

Hence, (b) and (c) are the correct answers.

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Application of Derivatives 4 Question 33

35. If $f (x)$ is a cubic polynomial which has local maximum at $x = - 1$ . If $f (2) = 18, f (1) = - 1$ and $f^{'} (x)$ has local minimum at $x = 0$ , then

Answer:

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Application of Derivatives 4 Question 33

35. If f(x) is a cubic polynomial which has local maximum at x=−1. If f(2)=18,f(1)=−1 and f′(x) has local minimum at x=0, then

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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35. If $f (x)$ is a cubic polynomial which has local maximum at $x = - 1$ . If $f (2) = 18, f (1) = - 1$ and $f^{'} (x)$ has local minimum at $x = 0$ , then