x = 5 sin t, y = 6 cos t, 0 < t < 2π

for what values is this concave up

Parametric equations: x = 5 sin t; y = 6cos t; 0 < t < 2 π.

Find the first derivative of y by x: y'_{x} = y'_{t} / x'_{t} = ( – 6sin t )/( 5 cos t) = (– 6/5) tan t.

Find the second derivative of y by x: y''_{xx} = (y'_{x} )'_{t} / x'_{t} = ( –6/5 cos^{–2} t) / 5 cos t= –6/25 cos^{–3} t

y''_{xx }= –6/25 cos^{–3} t > 0 if cos t < 0 . The curve is concave upward if t is in the interval (π/2 , 3π/2).

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