$A$ manufacturer sells $x$ items at a price of rupees $\left(6-\frac{x}{40}\right)$ each. The cost price of $x$ items is Rs. $\left(\frac{x}{5}+193\right)$. The maximum profit in Rs. is

$A$ manufacturer can sell $x$ items at a price of rupees $\left(5 - \frac{x}{100}\right)$ each. The cost price of $x$ items is Rs. $\left(\frac{x}{5} + 500\right)$. The number of items that the manufacturer should sell to earn the maximum profit is

$A$ tank with a rectangular base and rectangular sides,open at the top,is to be constructed such that its depth is $2 \ m$ and volume is $8 \ m^{3}$. If the cost of building the tank is Rs $70$ per square metre for the base and Rs $45$ per square metre for the sides,what is the minimum cost of the tank?

If $y = a \ln |x + 1| + b(x + 1)^2 + x$ has an extremum value of $4$ at $x = 0$,then the value of $(a, b)$ is:

Find the local maximum and local minimum values for the function $f(x) = \sin x - \cos x$ where $0 < x < 2\pi$.

The function $f(x) = x^5 - 5x^4 + 5x^3 - 1$ is: