How do you find the maximum, minimum and inflection points and concavity for the function #f(x) = 2x^2 + 6x + 5#?
2 Answers
This is a quadratic function.
Explanation:
A function
It opens upward and is concave up if
The function has a minimum if
The minimum or maximum occurs at the vertex which is at
(If you forget the veertex formula, use the derivative
See below.
Explanation:
This is just a standard quadratic. You do not need calculus to solve this problem. The coefficient of
So the function is convex ( concave up ) for
There are no inflection points, because the function is convex for the entire domain. Infection points only occur when the concavity changes.
Maximum value is
( this can be deduced by the fact that the function is concave up for entire domain )
To find the minimum value, we need to arrange the function in form:
Where a is the coefficient of
Bracket of the terms containing the variable:
Factor out the coefficient of
Add the square of half the coefficient of
Convert to the square of a binomial:
So minimum value
graph{y=2x^2+6x+5 [-5, 2, -5, 7]}