Interval in which the function $f(x) = 2x^3 - 3x^2 - 12x + 10$ is decreasing is :

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$. The value of $k$ is :

If $f(x) = x^4 - \dfrac{2x^3}{3}$, then $f(x)$ is

A. increasing in $\left(\dfrac{1}{2}, \infty\right)$

B. decreasing in $\left(\dfrac{1}{2}, \infty\right)$

C. increasing in $(-\infty, 0)$

D. decreasing in $(-\infty, 0) \cup \left(0, \dfrac{1}{2}\right)$

Choose the correct answer from the options given below:

The slope of the tangent to the curve $x = at^2$, $y = 2at$ at 't' is :

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$. The marginal cost when 10 items are produced is:

The maximum value of $f(x)=\dfrac{1}{3(x^2-x+1)}$ is :

The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is