The function S(x) = intlimits_0^x {sin left( | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $S(x) = \int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $.By the Fundamental Theorem of Calculus,$S'(x) = \sin \left( {\fr

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) Given $S(x) = \int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $.By the Fundamental Theorem of Calculus,$S'(x) = \sin \left( {\frac{{\pi {x^2}}}{2}} \right)$.Critical points occur where $S'(x) = 0$,so $\sin \left( {\frac{{\pi {x^2}}}{2}} \right) = 0$.This implies $\frac{{\pi {x^2}}}{2} = n\pi$ for some integer $n$,so $x^2 = 2n$.Given the interval $[1, 2.4]$,we have $1 \le x^2 \le 5.76$.For $n=1$,$x^2 = 2 \implies x = \sqrt{2} \approx 1.414$.For $n=2$,$x^2 = 4 \implies x = 2$.Now,find the second derivative: $S''(x) = \cos \left( {\frac{{\pi {x^2}}}{2}} \right) \cdot \pi x$.At $x = \sqrt{2}$,$S''(\sqrt{2}) = \cos(\pi) \cdot \pi \sqrt{2} = -\pi \sqrt{2} At $x = 2$,$S''(2) = \cos(2\pi) \cdot 2\pi = 2\pi > 0$,so $x = 2$ is a local minimum.Thus,the local minimum occurs at $x = 2$.

The function $S(x) = \int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $ has two critical points in the interval $[1, 2.4]$. One of the critical points is a local minimum and the other is a local maximum. The local minimum occurs at $x =$

Similar Questions

$x$ and $y$ are two positive integers such that $2x + 3y = 50$. If $x^2 y^3$ is maximum for $x = \alpha$ and $y = \beta$, then $\frac{\alpha}{2} + \frac{\beta}{5} =$

If the point of minima of the function $f(x) = 1 + a^2x - x^3$ satisfies the inequality $\frac{x^2 + x + 2}{x^2 + 5x + 6} < 0$,then $a$ must lie in the interval:

If $x$ and $y$ are two positive integers such that $x + 2y = 10$ and $x^2 y^3$ is maximum,then $x^2 + 2y^3 =$

The absolute maximum value of the function $f(x)=2 x^3-9 x^2+12 x+1$ on the interval $[0,2]$ is:

Let $f: R \rightarrow R$ be a function defined by $f(x) = (x - 3)^{n_{1}}(x - 5)^{n_{2}}$,where $n_{1}, n_{2} \in N$. Which of the following is $\text{NOT}$ true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module