Definite Integration Question 9
Question 9
- Let $f(x)=\int_{1}^{x} \sqrt{2-t^{2}} d t$. Then, the real roots of the equation $x^{2}-f^{\prime}(x)=0$ are (a) \pm 1 (b) $\pm \frac{1}{\sqrt{2}}$ (c) $\pm \frac{1}{2}$ (d) 0 and 1
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Answer:
Correct Answer: 9. (a)
Solution:
- Given, $f(x)=\int_{1}^{x} \sqrt{2-t^{2}} d t \Rightarrow f^{\prime}(x)=\sqrt{2-x^{2}}$
Also, $x^{2}-f^{\prime}(x)=0$
$$ \begin{array}{rlrl} & \therefore & x^{2}=\sqrt{2-x^{2}} \quad \Rightarrow \quad x^{4}=2-x^{2} \ & \Rightarrow & x^{4}+x^{2}-2 & =0 \Rightarrow x= \pm 1 \end{array} $$
