Applications of Derivatives Questions

A manufacturing firm sets the price per unit as $p(x)=400-\alpha x^2$, where $\alpha$ is a constant and $x$ is the units demanded. If the marginal revenue when 10 units were in demand was zero, then the value of $\alpha$ is:

A toy manufacturing firm has its variable cost $C(x)={\alpha }^{2}x(\beta \sqrt{x}+\alpha \beta )$, where $x$ is the number of toys produced. The cost of storage and other fixed expenses are $Rs.4500$. If the marginal cost is $40$ when $144$ toys have been produced and $\alpha \beta =1$, find the possible values of $\alpha$.

The function $f(x)={\tan }^{-1}x-x$ is monotonically decreasing in the set:

The function $f(x)={\sin }^{4}x+{\cos }^{4}x$ increases monotonically if: