Let $$f(x)=(sin(x))^3+ lambda(sin(x))^2$$ for every $x$ in domain of $(frac-π2,fracπ2)$ 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 $(3sinx + 2lambda)$ $ $ 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?

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

Solution

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

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 $f''(0)=2lambda.$ 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.