Application of DerivativeHard
Question
-If f(x) = sin3x + λ sin2 x ; -π/2 < x < π/2, then the interval in which l should lie in order that f(x) has exactly one minima and one maxima
Options
A.(-3/2, 3/2) - {0}
B.(-2/3, 2/3) - {0}
C.R
D.
Solution
f′(x) = sinx cos x (3 sin x + 2l)
f′(x) = 0
⇒ sin x = 0 or cos x = 0 or sin x =
⇒ x = 0 or sin x =
(as cos x = 0 is not possible).
If λ = 0 then f′(x) ≥ 0 ⇒ no extrema,
⇒ - 1 <
< 0 or 0 < 
< 1
⇒ 0 < λ < 3/2 or - 3/2 < λ < 0
f′(x) = 0
⇒ sin x = 0 or cos x = 0 or sin x =
⇒ x = 0 or sin x =
(as cos x = 0 is not possible).If λ = 0 then f′(x) ≥ 0 ⇒ no extrema,
⇒ - 1 <
< 0 or 0 < < 1⇒ 0 < λ < 3/2 or - 3/2 < λ < 0
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