Question

How do you determine the concavity of a quadratic function?

Answer 1

Hint:Concavity of a function is the rate of change of the function’s slope. As we know that the slope of a function is itself the rate of change of the function i.e The first derivative of the function is its slope, therefore, the first derivative of slope or we can say that the second derivative of the function defines the concavity of that function. So, to determine the concavity of a quadratic function, we need to find its second derivative.

Complete step by step solution:
(i)As we have to determine the concavity of a quadratic function, we must first know what concavity is.

So, concavity of any function is the rate of change of that function’s slope. It tells if the slope of the function is increasing or decreasing.

In simpler words, if the concavity of a function is positive, it depicts that the slope of the function is increasing and the function’s graph would be concave up.

While, if the concavity of a function is negative, it depicts that the slope is decreasing and the function’s graph is concave down.

Also, if the concavity of a function is zero, it depicts that the slope of the function is constant i.e., neither decreasing nor increasing. But we will look in this solution afterwards for how the concavity of a quadratic function can never be zero.

So, let us take the standard form of a quadratic function as $f(x)$
$f(x) = a{x^2} + bx + c$ [where $a \ne 0$]

(ii) First, we will calculate the first derivative of the function i.e., its slope. So,
$f'(x) = 2ax + b$

(iii) Now, for determining the concavity of the function $f(x)$, we will calculate its second derivative. So,
$f''(x) = 2a$

(iv) As we have got the concavity of the function $f(x)$ as $2a$, we can say that the sign of $f''

(v)$ directly depends on the sign of the coefficient $a$ as $2$ is a positive number, and if $a$ will be positive, $2a$ will be positive. Whereas, if $a$ will be negative, $2a$ will be negative.

Therefore, we can say that $a$ directly correlates with the concavity of the function, as if $a$ is positive,

$f''(x)$ will be positive and the function will be concave up, the same can be said for a negative value of
$a$, making $f''(x)$ negative resulting in the function being concave down.

Also $f''(x)$ can never be zero as for making $f''(x)$ zero, the value of $a$ should be zero and in a quadratic function, coefficient of ${x^2}$ can never be zero.

Hence, For a quadratic function $f(x) = a{x^2} + bx + c$,
If $a > 0$, then $f(x)$ is concave upward everywhere,
If $a < 0$, then $f(x)$ is concave downward everywhere.

Note: As we know that the graph of a quadratic function is a parabola, we can directly conclude that the sign of the coefficient of ${x^2}$ decides the concavity of the function as in a parabola $y = a{x^2} + bx + c$, if $a$ is positive, the parabola is open upwards and if $a$ is negative, the parabola is open downwards.