If x = 3tan(t) and y = 3sec(t), the value of d²y/dx² at t = π/4 is:

Given x = 3tan(t) and y = 3sec(t)
Differentiating x with respect to t, we get:
dx/dt = 3sec²(t)
Differentiating y with respect to t, we get:
dy/dt = 3sec(t)tan(t)
Now, dy/dx = (dy/dt) / (dx/dt) = (3sec(t)tan(t)) / (3sec²(t)) = tan(t)/sec(t) = sin(t)
Differentiating dy/dx with respect to t, we get:
d(dy/dx)/dt = cos(t)
Now, d²y/dx² = [d(dy/dx)/dt] / (dx/dt) = cos(t) / (3sec²(t)) = cos(t) / (3/cos²(t)) = cos³(t)/3
At t = π/4, cos(t) = cos(π/4) = 1/√2
Therefore, d²y/dx² at t = π/4 is:
(1/√2)³ / 3 = 1/(3 * 2√2) = 1/(6√2) = √2/12
This value does not match any of the given options. Let's re-examine the calculations.
We have dy/dx = sin(t)
Then d(dy/dx)/dt = cos(t)
d²y/dx² = (d(dy/dx)/dt) / (dx/dt) = cos(t) / (3sec²(t)) = cos³(t)/3
At t = π/4, cos(π/4) = 1/√2
d²y/dx² = (1/√2)³/3 = 1/(3 * 2√2) = √2/12
Let's check the calculation of dy/dx again:
dy/dx = (dy/dt)/(dx/dt) = (3sec(t)tan(t))/(3sec²(t)) = tan(t)/sec(t) = sin(t)
d/dt(dy/dx) = cos(t)
d²y/dx² = [d/dt(dy/dx)] / (dx/dt) = cos(t) / (3sec²(t)) = cos³(t)/3
At t = π/4, cos(π/4) = 1/√2
d²y/dx² = (1/√2)³/3 = 1/(3(2√2)) = √2/12 ≈ 0.0589
There seems to be an error in the provided solution. The correct calculation is shown above. None of the given options match the calculated value.