Increasing and Decreasing Functions

If the function f(x) = x³ − 9kx² + 27x + 30 is increasing on R, then

Function f(x) = 2x³ − 9x² + 12x + 29 is monotonically decreasing when

Find the value of x, such that f(x) is increasing function.

f(x) = x² + 2x - 5 

Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?

If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______

Let f(x) = x³ + ax² + bx + 5 sin²x be an increasing function on the set R. Then, a and b satisfy.

Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.

Find `dy/dx,if e^x+e^y=e^(x-y)`