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In plotting graphics we use the first derivative to find critical points and in which intervals the function grows and becomes smaller. We can insert the critical points in the second derivative to see if it is convex or concave at these points.

But sometimes the second derivative is really hard to find. It is important because it gives us additional information, like inflex points, which help us make our graph more precise.

Is there any approach if the second derivative is hard to find?

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  • $\begingroup$ When is a derivative of a known function hard to find? Derivatives are generally easy to find, even if they may be hard to set equal to zero. Do you have an example? $\endgroup$ – 6005 Jan 1 '15 at 19:20
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You do not limit the answers to purely analytic means, so here is a practical answer. (Note that the current title with the comment "by hand" was not in the original title, and the comment was added by someone other than the OP, so this answer is relevant to the original question.)

A not-perfect-but-not-to-be-despised answer is to use a numerical graphing program, web page, or graphing calculator. These require care, since graphing errors are possible, but they definitely help in understanding the graph of a function.

In my calculus class, I encourage students to solve a problem analytically if possible, then to check the answer graphically or numerically. Therefore, I require the use of a graphing calculator. If this were a college-level class, I would also encourage graphing programs.

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To plot a function $f$, I generally find it sufficient to:

  1. Pick several $x$-values, including the $x$-values where $f(x) = 0$ and where $f'(x) = 0$.

  2. Plot the point $(x, f(x))$ for each $x$-value. Draw a light tangent line at each $x$-value indicating the value of $f'(x)$ at that point.

  3. Use the points and tangent lines to sketch the curve.

For more accuracy, use more $x$-values. I generally don't find the second derivative necessary. For example, instead of using the so-called "second derivative test" you can just check if the first derivative changes sign or not, and if so, in which direction it changes sign.

Of course, in practice, using mathematical software is going to be quicker and more precise anyway.

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