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

Question

Let $$f(x)=(sin(x))^3+ \lambda(sin(x))^2$$ for every $x$ in domain of $(\frac{-π}{2},\frac{π}{2})$ has no point of local maxima or minima, then find the value of $\lambda$ for the given condition to follow.

My process:

(1): found $f'(x)$ and equated with zero to find that it only matters by $(3sinx + 2\lambda)$ $ $ and $sinx$ to have change in the sign of derivative around zero.So the $\lambda$ becomes 0 and 0 becomes the point of inflection. Is my process wrong?

Answer 1

Solution

We have $$f'(x)=3\sin^2 x\cos x+2\lambda\sin x \cos x=\sin x \cos x(3\sin x+2\lambda),$$ and $$f''(x)=2\cos^2 x(3\sin x+\lambda)-\sin^2 x(2\lambda+3\sin x).$$

Notice that, whatever $\lambda$ is, $f'(0)=0.$ Thus, we at least need $f''(0)=0.$ Otherwise, $f(x)$ reaches its local extremum value at $x=0$. But $f''(0)=2\lambda.$ Hence $\lambda=0$, which has only one possible value. Now, we may verify that $f(x)=\sin^3 x$ could satisfy the conditions we supposed.

Answer 2

Yes, your process is correct, the only $\lambda$ for which there is no maxima or minima in the interval $(-\pi/2,\pi/2)$ is $\lambda = 0$.