Eduprobe
Question:
Let a, b ∈ R and f: R → R be defined by f(x) = acos(|x³ - x|) + b|x|sin(|x³ + x|). Then f is
Differentiable at x=0 if a =0 and b = 1
Differentiable at x=1 if a =1 and b = 0
NOT Differentiable at x=0 if a =1 and b = 0
NOT Differentiable at x=1 if a =1 and b = 1
Solution:
f(x) = acos(|x³ - x|) + b|x|sin(|x³ + x|)
For (A) → If a = 0, b = 1, f(x) = |x|sin(|x³ + x|) → f(x) = xsin(x³ + x), ∀x ∈ R
Hence f(x) is differentiable.
For (B), (C) → If a = 1, b = 0, f(x) = cos(|x³ - x|) → f(x) = cos(x³ - x), which is differentiable at x = 1 and x = 0
For (D) → If a = 1, b = 1, f(x) = cos(x³ - x) + |x|sin(|x³ + x|) = cos(x³ - x) + xsin(x³ + x), which is differentiable at x = 1