Parametric equations: x = e^{t}; y = t e^{–t}

Find the first derivative of y by x: y'_{x} = y'_{t} / x'_{t} = ( e^{–t } – te^{–t }) / e^{t }= e^{–2t } – te^{–2t }

Find the second derivative of y by x: y''_{xx} = (y'_{x} )'_{t} / x'_{t} = ( –2e^{–2t } – e^{–2t }+2te^{–2t }) / e^{t }=

(–3e^{–2t }+ 2te^{–2t })/e^{t }= –3e^{–3t }+ 2te^{–3t }= (2t – 3)e^{–3t}

y''_{xx }= (2t – 3)e^{–3t }> 0 if t > 3/2 = 1.5 . The curve is concave upward if t is in the interval (1.5, +∞).

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