The set of all real values of λ for which the function f(x) = (1 - cos²x) ⋅ (λ + sinx), x ∈ (-π/2, π/2), has exactly one maxima and exactly one minima, is
The set of all real values of λ for which the function f(x) = (1 - cos²x) ⋅ (λ + sinx), x ∈ (-π/2, π/2), has exactly one maxima and exactly one minima, is
Option 1 -
(-3/2, 3/2) - {0}
Option 2 -
(-3/2, 3/2)
Option 3 -
(-1/2, 1/2)
Option 4 -
(-1/2, 1/2) - {0}
Asked by Shiksha User
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1 Answer
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Correct Option - 4
Detailed Solution:f (x) = sin²x (λ + sinx)
f' (x) = 2sinxcosx (λ + sinx) + sin²x (cosx) = sinxcosx (2λ + 3sinx)
For extrema, f' (x) = 0
sinx = 0, cosx = 0, or sinx = -2λ/3
For more than 2 points of extrema in the interval, sinx = -2λ/3 must have solutions other than where sinx=0 or cosx=0.
-1 < -2λ/3 < 1 and -2λ/3 ≠ 0
This gives λ ∈ (-3/2, 3/2) - {0}
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