Continuity and DifferentiabilityHard

Question

The slope(s) of common tangent(s) to the curves y = e-x & y = e-x sin x can be -

Options

A.-e-π/2
B.-e-π
C.π/2
D.1

Solution

y = e-x        &    y = e-xsinx
y′ = -e-x  ...(i)    & y′ = -e-x(sin x - cos x) ...(ii)
equating (i) & (ii)
e-x(1 - sin x + cos x) = 0
e-x ≠ 0    ⇒1 - sin x + cos x = 0
⇒ 2cos2 = 2 sin cos
⇒  2 cos = 0 ⇒ x = , π
slope can be -e-π/2    &    -e-π.

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