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in Continuity and Differentiability by (29.1k points)
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Find the set of values of ‘b’ for which f(x) = b(x + cos x) + 4 is decreasing on R.

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f(x) = b (x + Cos x) + 4

f’(x) = b (1 – Sin x) + 0

f’(x) = b (1 – Sin x)

\(\because\) f(x) is decreasing on R.

\(\therefore\) f’(x) < 0

b (1 – Sin x) < 0

\(\because\) Sin x ≤ 1

1 – Sin x ≥ 0

\(\because\) b (1 – Sin x) < 0 & 1 – Sin x ≥ 0

\(\therefore\) b < 0

\(\therefore\) b ∈ (-∞, 0)

Hence the required set of values is b ∈ (-∞, 0).

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