If x = at^2 and y = 4at, then the value of d^2y/dx^2 at t = 1/a, is Option 1: a Option 2: -a^2 Option 3: -a Option 4: 1/a

Answer: -a^2. Solution: dy/dx=(dy/dt)/(dx/dt)=4a/2at=2/t. Then d^2y/dx^2=d/dt(2/t)dt/dx=(-2/t^2)1/2at=-1/at^3. At t=1/a, this is -1/(a*(1/a^3))=-a^2.

Calculus

Continuity & Differentiability

Medium

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If $x = at^2$ and $y = 4at$ , then the value of $\frac{d^2y}{dx^2}$ at $t = 1/a$ , is

- $a^2$

-a

1/a

✅ Correct Option: 2

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{4a}{2at}=\frac{2}{t}$ .

Then $\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{2}{t}\right)\cdot\frac{dt}{dx}=\left(-\frac{2}{t^2}\right)\cdot\frac{1}{2at}=-\frac{1}{at^3}$ .

At $t=1/a$ , this is $-\frac{1}{a\cdot(1/a^3)}=-a^2$ .

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If x=at2x = at^2x=at2 and y=4aty = 4aty=4at, then the value of d2ydx2\frac{d^2y}{dx^2}dx2d2y​ at t=1/at = 1/at=1/a, is

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If $x = at^2$ and $y = 4at$ , then the value of $\frac{d^2y}{dx^2}$ at $t = 1/a$ , is