SATHEE: Limit Continuity and Differentiability 7 Question 26

Limit Continuity and Differentiability 7 Question 26

26. For every twice differentiable function $f : R \to [- 2, 2]$ with $(f (0))^{2} + {(f^{'} (0))}^{2} = 85$ , which of the following statement(s) is (are) TRUE?

(2018 Adv.)

(a) There exist $r, s \in R$ , where $r < s$ , such that $f$ is one-one on the open interval $(r, s)$

(b) There exists $x_{0} \in (- 4, 0)$ such that $| f^{'} (x_{0}) | \leq 1$

(d) There exists $α \in (- 4, 4)$ such that $f (α) + f^{''} (α) = 0$ and $f^{'} (α) \neq 0$

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Answer:

Correct Answer: 26. (b, d)

Solution:

Given, $F (x) = | \begin{array}{lll} f_{1} (x) & f_{2} (x) & f_{3} (x) g_{1} (x) & g_{2} (x) & g_{3} (x) h_{1} (x) & h_{2} (x) & h_{3} (x) \end{array} |$

$∴ F^{'} (x) = | \begin{array}{ccc} f_{1}^{'} (x) & f_{2}^{'} (x) & f_{3}^{'} (x) g_{1} (x) & g_{2} (x) & g_{3} (x) h_{1} (x) & h_{2} (x) & h_{3} (x) \end{array} |$

$+ | \begin{array}{ccc} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ g_{1}^{'} (x) & g_{2}^{'} (x) & g_{3}^{'} (x) \\ h_{1} (x) & h_{2} (x) & h_{3} (x) \end{array} | + | \begin{array}{ccc} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ g_{1} (x) & g_{2} (x) & g_{3} (x) \\ h_{1}^{'} (x) & h_{2}^{'} (x) & h_{3}^{'} (x) \end{array} |$

$\Rightarrow F^{'} (a) = 0 + 0 + 0 = 0$

$[∵ f_{r} (a) = g_{r} (a) = h_{r} (a); r = 1, 2, 3]$

Limit Continuity and Differentiability 7 Question 26

26. For every twice differentiable function $f : R \to [- 2, 2]$ with $(f (0))^{2} + {(f^{'} (0))}^{2} = 85$ , which of the following statement(s) is (are) TRUE?

Answer:

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Limit Continuity and Differentiability 7 Question 26

26. For every twice differentiable function f:R→[−2,2] with (f(0))2+(f′(0))2=85, which of the following statement(s) is (are) TRUE?

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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26. For every twice differentiable function $f : R \to [- 2, 2]$ with $(f (0))^{2} + {(f^{'} (0))}^{2} = 85$ , which of the following statement(s) is (are) TRUE?