Question

Investigate the given functions for monotony and extrema `y=2x^2+3x+1`?

Answer 1

Consider derivatives

Explanation:

Consider the first derivative:
`dy/dx=4x+3`

Set this equal to zero to find extrema and points of inflection:
`4x+3=0rArrx=-3/4`

Consider the second derivative at this point:
`(d^2y)/(dx^2)=4`
It is constant for all `x`, so `>0` at the zero first derivative point above. Thus this point is a minimum of the function.

"Monotony" refers to a lack of turning points - a monotonic function is one that either increases or decreases uniformly through its range. This function divides into monotonically decreasing and monotonically increasing pieces, one of each, which change over at the single turning point.