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Project: CC9.6
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def cal():
    C = 100
    R = 50
    a = 5
    b = 10
    T = var('T')
    g(T) = (C - R * (b - T)/(b - a) * e ^ (-2 * T / (a + b))) / (T - (T - a)^2 / (2 * (b - a)))
    gprime = derivative(g,T)
    P = plot(g,(T,0,50),color="blue")
    Q = plot(gprime,(T,0,50),color="red")
    i = 0
    mins = 1000000000000000000
    mini = -1
    while i <= 10000:
        point = g(i).n(30)
        if point < mins:
            mins = point
            mini = i
        i += 1
    i_left = mini - 1
    i_right = mini + 1
    i = i_left
    # show(i_left,i_right)
    mins = 1000000000000000000
    mini = -1
    while i <= i_right:
        point = g(i).n(30)
        if point < mins:
            mins = point
            mini = i
        i += 0.00001
    show("min_g = ", mins)
    show("min_T = ", mini)
    show(g)
    show(P)
    show(gprime)
    show(Q)
cal()
min_g = −1.0380293×107\displaystyle -1.0380293 \times 10^{7}
min_T = 18.6602599999750\displaystyle 18.6602599999750
T ↦ −100 ((T−10)e(−215 T)+10)(T−5)2−10 T\displaystyle T \ {\mapsto}\ -\frac{100 \, {\left({\left(T - 10\right)} e^{\left(-\frac{2}{15} \, T\right)} + 10\right)}}{{\left(T - 5\right)}^{2} - 10 \, T}
T ↦ 20 (2 (T−10)e(−215 T)−15 e(−215 T))3 ((T−5)2−10 T)+200 ((T−10)e(−215 T)+10)(T−10)((T−5)2−10 T)2\displaystyle T \ {\mapsto}\ \frac{20 \, {\left(2 \, {\left(T - 10\right)} e^{\left(-\frac{2}{15} \, T\right)} - 15 \, e^{\left(-\frac{2}{15} \, T\right)}\right)}}{3 \, {\left({\left(T - 5\right)}^{2} - 10 \, T\right)}} + \frac{200 \, {\left({\left(T - 10\right)} e^{\left(-\frac{2}{15} \, T\right)} + 10\right)} {\left(T - 10\right)}}{{\left({\left(T - 5\right)}^{2} - 10 \, T\right)}^{2}}