Find all critical points of the function g(theta)=sin^(2)(6theta). E

Question

Find all critical points of the function  g(theta)=sin^(2)(6theta). E

Bytelearn · Accepted Answer

Understand critical points: Understand what critical points are. Critical points of a function occur where the derivative is zero or undefined. To find the critical points of g(theta) = sin^2(6theta), we need to find the derivative g'(theta) and solve for theta where g'(theta) = 0 or where g'(theta) is undefined.Find derivative of g(theta): Find the derivative of g(theta) = sin^2(6theta). Using the chain rule, the derivative of sin^2(u) with respect to u is 2sin(u)cos(u). Here, u = 6theta, so we need to use the chain rule again to differentiate u with respect to theta, which gives us 6. Therefore, the derivative of g(theta) with respect to theta is g'(theta) = 2sin(6theta)cos(6theta) * 6.Simplify the derivative: Simplify the derivative. We can use the double-angle formula for sine, sin(2u) = 2sin(u)cos(u), to simplify g'(theta). This gives us g'(theta) = 6sin(2 * 6theta) = 6sin(12theta).Set derivative equal to zero: Set the derivative equal to zero to find critical points. To find the critical points, we need to solve 6sin(12theta) = 0. Since the coefficient 6 is not zero, we can divide both sides by 6 to get sin(12theta) = 0.Solve sin(12theta) = 0: Solve the equation sin(12theta) = 0. The sine function is zero at integer multiples of pi. Therefore, 12theta = n*pi, where n is an integer. To find theta, we divide both sides by 12 to get theta = n*pi/12, where n is an integer.Determine interval for theta: Determine the interval for theta. Since theta is an angle, we typically consider its values in a specific interval, such as [0, 2pi) for one full rotation. However, the problem does not specify an interval, so we will provide the general solution theta = n*pi/12, where n is an integer.Check for undefined points: Check for any points where the derivative is undefined. The derivative g'(theta) = 6sin(12theta) is defined for all real numbers, so there are no critical points where the derivative is undefined.

Find all critical points of the function $g(\theta)=\sin^{2}(6\theta).$

Full solution

More problems from Find derivatives of logarithmic functions

Related topics

Find all critical points of the function g(θ)=sin⁡2(6θ).g(\theta)=\sin^{2}(6\theta).g(θ)=sin2(6θ).

Find all critical points of the function $g(\theta)=\sin^{2}(6\theta).$