For all points on the curve x^2 - siny=y+4 where y=pi/2, is the curve concave up or concave down? Why?
if y = π/2, then
x^2 - 1 = π/2 + 4
x = ±√(π/2 + 5)
so, since
y' = x sec^2(y/2)
y" = 2x^2 sin(y/2) sec^5(y/2)
just check the sign of y" at the given values of x.
To determine whether the curve x^2 - siny = y + 4 is concave up or concave down for all points on the curve where y = pi/2, you need to find the second derivative of the equation and evaluate it at the given value of y.
First, let's find the first derivative of the equation with respect to x:
d/dx (x^2 - siny) = d/dx (y + 4)
Differentiating x^2 gives us: 2x
Differentiating -siny gives us: -cosy * dy/dx
Since the derivative of y with respect to x is dy/dx, this becomes: -cosy * dy/dx
Therefore, the first derivative of the equation is: 2x - cosy * dy/dx
To find the second derivative, we need to differentiate the first derivative with respect to x:
d/dx (2x - cosy * dy/dx) = d^2y/dx^2
Differentiating 2x gives us: 2
Differentiating -cosy * dy/dx gives us: sin y * (dy/dx)^2 - cosy * d^2y/dx^2
Therefore, the second derivative is: 2 + sin y * (dy/dx)^2 - cosy * d^2y/dx^2
Substituting y = pi/2, we get: 2 + sin(pi/2) * (dy/dx)^2 - cos(pi/2) * d^2y/dx^2
Simplifying, sin(pi/2) equals 1 and cos(pi/2) equals 0, so the equation becomes: 2 + (dy/dx)^2
Since the second derivative, 2 + (dy/dx)^2, is always positive for any value of dy/dx, regardless of the value of x, it indicates that the curve is concave up for all points on the curve where y = pi/2.
Therefore, the curve x^2 - siny = y + 4 is concave up.