Question

What is the concavity of a linear function?

Answer 1

Hint: Concavity of a function is defined as 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 concave of the function. We need to find out the concavity of a linear function.

Complete step-by-step solution:
As we have to determine the concavity of a linear function, we must first know the 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 simple terms, 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.
If the concavity of a function is negative, it depicts that the slope of the function is decreasing and the function’s graph would be concave down.
Also, if the concavity of a function is zero, it depicts that the slope of the function is constant. i.e., neither increasing nor decreasing.
Let us solve the question,
A linear is in the form \[f\left( x \right)=mx+b\], where m is the slope, x is the variable and b is the y-intercept.
We can find the concavity of a function by finding its second derivative \[\left( f''\left( x \right) \right)\] and where it is equal to zero.
 Let’s do the problem
\[f\left( x \right)=mx+b\]
Differentiating the above linear function (first derivative)
\[\Rightarrow f'\left( x \right)=m\cdot 1\cdot {{x}^{1-1}}+0\]
Evaluating the above function
\[\begin{align}
  & \Rightarrow f'\left( x \right)=m\cdot 1 \\
 & \Rightarrow f'\left( x \right)=m \\
\end{align}\]
Second derivative of a linear function,
\[\therefore \left( f''\left( x \right) \right)=0\].
Therefore, the second derivative of a linear function is zero, there is no concavity.

Note: As, we know that linear functions are a straight line, therefore, there is no point of concavity on the graphs of linear functions. The concavity of quadratic function depends on the sign of the coefficient of \[{{x}^{2}}\]in a parabola \[y=a{{x}^{2}}+bx+c\], if \[a\]is positive, the parabola is open upwards and \[a\]is negative, the parabola is open downwards.