Limit Continuity and Differentiability 7 Question 26
26. For every twice differentiable function $f: R \rightarrow[-2,2]$ with $(f(0))^{2}+\left(f^{\prime}(0)\right)^{2}=85$, 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 $\left|f^{\prime}\left(x _0\right)\right| \leq 1$
(c) $\lim _{x \rightarrow \infty} f(x)=1$
(d) There exists $\alpha \in(-4,4)$ such that $f(\alpha)+f^{\prime \prime}(\alpha)=0$ and $f^{\prime}(\alpha) \neq 0$
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Answer:
Correct Answer: 26. (b, d)
Solution:
- Given, $F(x)=\left|\begin{array}{lll}f _1(x) & f _2(x) & f _3(x) \ g _1(x) & g _2(x) & g _3(x) \ h _1(x) & h _2(x) & h _3(x)\end{array}\right|$
$\therefore \quad F^{\prime}(x)=\left|\begin{array}{ccc}f _1{ }^{\prime}(x) & f _2{ }^{\prime}(x) & f _3{ }^{\prime}(x) \ g _1(x) & g _2(x) & g _3(x) \ h _1(x) & h _2(x) & h _3(x)\end{array}\right|$
$$ +\left|\begin{array}{ccc} f _1(x) & f _2(x) & f _3(x) \\ g _1^{\prime}(x) & g _2^{\prime}(x) & g _3^{\prime}(x) \\ h _1(x) & h _2(x) & h _3(x) \end{array}\right|+\left|\begin{array}{ccc} f _1(x) & f _2(x) & f _3(x) \\ g _1(x) & g _2(x) & g _3(x) \\ h _1^{\prime}(x) & h _2^{\prime}(x) & h _3^{\prime}(x) \end{array}\right| $$
$$ \Rightarrow \quad F^{\prime}(a)=0+0+0=0 $$
$$ \left[\because f _r(a)=g _r(a)=h _r(a) ; r=1,2,3\right] $$
