Find λ such that f(x) has no point of local maxima or minima in |x|&lt;π2 Let f(x)=(sin⁡(x))3+λ(sin⁡(x))2 for every x in domain of (−π2,π2) has no point of local maxima or minima, then find the value of λ for the given condition to follow. 1): found f′(x) and equated with zero to find that it only matters by (3sin⁡x+2λ) and sinx to have change in the sign of derivative around zero.So the λ becomes 0 and 0 becomes the point of inflection.

Question

Find λ such that f(x) has no point of local maxima or minima in |x|&amp;lt;π2  Let  f(x)=(sin⁡(x))3+λ(sin⁡(x))2  for every x in domain of (−π2,π2) has no point of local maxima or minima, then find the value of λ for the given condition to follow.  1): found f′(x) and equated with zero to find that it only matters by (3sin⁡x+2λ)  and sinx to have change in the sign of derivative around zero.So the λ becomes 0 and 0 becomes the point of inflection.

Micheal Hensley · Accepted Answer

Solution We have f′(x)=3sin2⁡xcos⁡x+2λsin⁡xcos⁡x=sin⁡xcos⁡x(3sin⁡x+2λ),andf″(x)=2cos2⁡x(3sin⁡x+λ)−sin2⁡x(2λ+3sin⁡x).Notice that, whatever λ is, f′(0)=0. Thus, we at least need f′′(0)=0. Otherwise, f(x) reaches its local extremum value at x=0. But f″(0)=2λ. Hence λ=0, which has only one possible value. Now, we may verify that f(x)=sin3x could satisfy the conditions we supposed.

Find \lambda such that f(x) has no point of local maxima or minima in |x|< \frac{\pi}{2}

Answered question

Answer & Explanation

New Questions in Precalculus