微分・積分～同志社・理系全学部・2019数学Ⅱ

(1) √2 (1/√2 sin x + 1/√2 cos x) =0,   sin (x + π/4) = 0     ∴ x = 3π/4, 7π/4

(2) C: y = exp(-x)   y'= -exp(-x)

     D: y = a sin x    y'= a cos x

     E: y = b sin x    y' = b cos x

     C, D は共有点Pで共通の接線　⇔　exp(-x) = a sin x　➀,  -exp(-x) = a cos x　②

　➀②を辺々足して　0 = a (sin x + cos x)    ∴ x = 3π/4, 7π/4

    ①に代入　exp(-3π/4) = 1/√2 a 　∴ a = √2 exp (-3π/4),   p = 3π/4

　同様に　　exp(-7π/4) = -1/√2 b 　∴ b = -√2 exp (-7π/4),  q = 7π/4