What are the critical points for the plane curve defined by the equations x(t)=-sin(3t),y(t)=5t, and 0

Question

What are the critical points for the plane curve defined by the equations x(t)=-sin(3t),y(t)=5t, and 0 <= t < pi ? Write your answer as a list of values of t, separated by commas. For example, if you found t=1 or t=2, you would enter 1,2 .

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Find x'(t): To find the critical points of the plane curve, we need to find the values of t where the derivatives of x(t) and y(t) are both zero or do not exist. Let's start by finding the derivative of x(t) with respect to t. x(t) = -sin(3t) dx/dt = -cos(3t) * 3Find y'(t): Now, let's find the derivative of y(t) with respect to t. y(t) = 5t dy/dt = 5No critical points for y(t): Since dy/dt = 5 is never zero and does not depend on t, there are no critical points for y(t) based on its derivative. Therefore, we only need to consider the critical points for x(t) based on dx/dt.Set x'(t) to zero: We set the derivative of x(t) equal to zero to find the critical points for x(t). 0 = -cos(3t) * 3 This implies that cos(3t) must be equal to 0.Solve for cos(3t)=0: The cosine function is zero at odd multiples of π/2. Therefore, we need to find the values of t such that 3t is an odd multiple of π/2 within the given interval 0 <= t < π.Find values of t: Let's solve for t: 3t = (2n+1)π/2, where n is an integer. t = (2n+1)π/6Solve for t: We need to find the values of n such that t is in the interval 0 <= t < π. Let's start with n=0 and increase n to find all possible values of t. For n=0: t = π/6 For n=1: t = 3π/6 = π/2 For n=2: t = 5π/6Find possible values of t: If we increase n to 3, we get t = 7π/6, which is outside the interval 0 <= t < π. Therefore, we only consider n=0, n=1, and n=2.Consider n=0,1,2: The critical points for the plane curve defined by the equations x(t)=-sin(3t) and y(t)=5t are at t=π/6, t=π/2, and t=5π/6.

What are the critical points for the plane curve defined by the equations $x(t)=-\sin (3 t), y(t)=5 t$ , and 0 \leq t<\pi ? Write your answer as a list of values of $t$ , separated by commas. For example, if you found $t=1$ or $t=2$ , you would enter $1$ , $2$ .

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