Application of differentiation for finding maxima and minima

Generally there are many tests to check maxima or minima of a function - most famous being first derivative test and second derivative test.

My question deals with second derivative test : We find first derivative check where it is zero (simply its algorithm) , and then second : if latter comes to be negative we say it to be maxima and vice versa, if zero then higher order derivatives and so on.

I was thinking about the situation geometrically. First derivative deals with slope of tangent : it is positive if function is increasing and vice versa.

Then what is significance of signs in higher derivatives of the situation. Do second derivative denote slope of tangent? Well not really. What do you think?

• second derivative helps to determine whether the graph is concave or convex on a particular interval which helps you to distinguish between max and min – Vasya Sep 5 '17 at 15:19
• google.com/… – Hans Lundmark Sep 5 '17 at 16:28

Second derivative denotes concavity of the underlying curve. $f''(x)>0$ means $f$ at $x$ is concave up (like $y=x^2$ for example) and $f''(x)<0$ means concave down. Finally, $f''(x)=0$ is an inflection point.