Why is it necessary to take the 2nd derivative to determine concavity?
I'm having trouble understanding why you need the second derivative to determine concavity.
For example, if I have the equation:
$y = -4x^2 + 24x + 42$
$y' = -8x +24$
I know from the first derivative alone that the slope is -8 from what you learn from $y=mx+b$. So what's the point in taking the second derivative? Isn't taking the slope of the slope (second derivative) redundant at this point?
For context: I have read / listened to explanations online. And I understand the explanations about getting the first derivative for the slope. But then the explanation says something along the lines of, "So it follows that the second derivative will give us what we need for concavity. If the slope is greater than 0,...If the slope is less than 0,..."
But if the first derivative is a tangent line (straight line) then what are we taking the slope for again? I thought taking the derivative could be used for straight lines, but was specifically useful for non-linear graphs. Otherwise, if it's linear, we could just use $y=mx+b$ to determine the slope.
Or am I oversimplified this? And the real point is that if you have higher order equations, you can differentiate until you have no variables and that gives you the slope?
Note: I did see this question, but I'm still confused. Concavity & Second Derivative
$\endgroup$ 44 Answers
$\begingroup$The value of the derivative is the slope of the tangent line. If the slope is positive the function is increasing, if negative the function is decreasing.
When you ask about concavity you are asking about how the slope is changing. If the slope is increasing then the curve is getting steeper, so "bending up" or "cap shaped". That means (usually) that it lies above the tangent line. If the slope is decreasing then the curve is "bending down" so it will lie beneath the tangent line. The easiest functions to thing about are $f(x) = x^2$ which is cap shaped, always above its tangents, and $f(x) = -x^2$ .
Since derivatives measure rates of change, one way to see whether the derivative itself is increasing or decreasing is to find its derivative: the second derivative of the original function. For the parabolas in the preceding paragraph, the first has constant second derivative $2$, which means the slope is increasing at that constant rate.
$\endgroup$ 1 $\begingroup$To emphasis a point that is not emphasized in other answers: using the 2nd derivative is not necessary to determine concavity, because there are alternate methods to determine concavity.
Take for example $f(x)=x^4$. You can easily verify that $f''(0)=0$, so the 2nd derivative test won't even work here. Nonetheless it's graph is concave upward at $x=0$, which you can observe by using a graphing calculator. There is an alternate method for verifying upward concavity of $f(x)=x^4$, as explained in other answers: verify directly that the first derivative $f'(x)=4x^3$ is an increasing function.
But this "1st derivative" test for concavity (i.e. $f'(x)$ is an increasing function) method is harder to apply than just using the 2nd derivative test (i.e. $f''(x)$ is a positive function), assuming that the 2nd derivative test works.
So the real point is that using the 2nd derivative test to verify concavity is very useful:
- It is very simple to use;
- Cases where it does not work are rare (e.g. $f(x)=x^4$);
- Alternative methods are harder to apply.
The first derivative gives you the slope of a line which is tangent to the graph at a point $x$.
The second derivative is a measure of how that slope changes as we vary $x$.
For example, consider the function $f(x)=x^2$. At $x=-1$, the slope of the tangent line is $-2$. At $x=0$, the slope of the tangent line is $0$. The fact that the slope of the tangent line increases with increasing $x$, is equivalent to saying that $f"(x)>0$, which is equivalent to saying that the graph is concave up
$\endgroup$ $\begingroup$Yes, in quadratics, the $x^2$ terms tells you if it concave; however, the method can be more useful with more complex functions. If, at a point, the second derivate is zero and the second derivate is positive you know (at least around that point) its not concave downwards and if its positive, your know it is, no matter if it's a quadratic or some two-line-trig-function-mess. Well, this wont work for some wired functions though but this should give you an idea of why.
$\endgroup$
