Let f(x) be a function such that f^ (x)=g(x) and f^ (x)=-f(x) . Let h(x)=\ f(x)\ ^2+\ g(x)\ ^2 . Then consider the following statements : 1. h^ (3)=0 2. h(1)=h(2) Which of the statements given above is/are correct?

Correct answer: C. Both 1 and 2 aligned & h(x)=\ f(x)\ ^2+\ g(x)\ ^2 & =\ f(x)\ ^2+\ f^ (x)\ ^2 & h^ (x)=2 f(x) · f^ (x)+2 f^ (x) f^ (x) & =2 f(x) f^ (x)-2 f(x) f^ (x)=0 & h' (x)=0 & h^ (3)=0 aligned Statement 1 is correct. aligned & h^ (x)=0 h(x)= constant function & h(1)=f(2) aligned Statement 2 is correct.

NDA 2022

Exam: NDA
Year: 2022
Subject: Mathematics
Chapter: Application of Derivatives

NDA 2022 · Mathematics · Application of Derivatives

Let f(x) be a function such that f^ (x)=g(x) and f^ (x)=-f(x) . Let...

Let

f(x)

be a function such that

f^{\prime}(x)=g(x)

and

f^{\prime \prime}(x)=-f(x)

. Let

h(x)=\{f(x)\}^2+\{g(x)\}^2

. Then consider the following statements : 1.

h^{\prime}(3)=0 2.

h(1)=h(2)

Which of the statements given above is/are correct?

Options

A1 only
B2 only
CBoth 1 and 2
DNeither 1 nor 2

Correct answer

C. Both 1 and 2

Step-by-step solution

\begin{aligned} & h(x)=\{f(x)\}^2+\{g(x)\}^2 \\ & =\{f(x)\}^2+\left\{f^{\prime}(x)\right\}^2 \\ & h^{\prime}(x)=2 f(x) \cdot f^{\prime}(x)+2 f^{\prime}(x) f^{\prime \prime}(x) \\ & =2 f(x) f^{\prime}(x)-2 f(x) f^{\prime}(x)=0 \\ & \because \quad \text{ h' }(x)=0 \\ & \therefore \quad h^{\prime}(3)=0 \end{aligned}

Statement 1 is correct.

\begin{aligned} & h^{\prime}(x)=0 \Rightarrow h(x)=\text{ constant function } \\ & \therefore \quad h(1)=f(2) \end{aligned}

Statement 2 is correct.

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