how can I find the concavity of the tangent line of the function?

Answer 1

I assume you want to calculate the curvature k of the graph of y = f(x) in a given point (x,y).

This is by definition 1/R, where R is the radius of the circle that approximates the curve best in the given point.

You can find k from the first and second derivatives, f''(x) and f'(x):

k = f''(x) / [1 + f'(x)^2]^(3/2)

For instance, if f(x) = ax^2 + bx + c, the curvature for arbitrary x is found as follows.

f'(x) = 2ax + b
f''(x) = 2a

k = 2a / [1 + (2ax + b)^2]^{3/2}

In the vertex of the parabola, 2ax + b = 0 and the formula simplifies to k = 2a.

Answer 2

A line is neither concave up nor concave down.
y = mx + b
y' = m
y'' = 0