I need help with solving this problem. The function , where x is any real number

Question

I need help with solving this problem.

The function , where x is any real number, is of fundamental importance in science and engineering. (The function is called the error function.) While we know that B exists, we must approximate its values numerically. Nevertheless, we can determine many of the properties of this function using our knowledge of derivatives and the Fundamental Theorems. Find B'(x) and show that B is increasing on all of its domain (- infinity, infinity). Show that the graph of B has exactly one inflection point, and then find that inflection point. Explain why B has an inverse B-1 and then find (using a calculator or computer) the value B-1(0.66). Round your answer to two decimal places.

Explanation / Answer

a) B'(x) = exp(-x^2)

Now we can see since it is always positive so B is an increasing function in its domain.

b) inflextion B''(x) = -2x*exp(-x^2) which changes sign only at x=0. So only one inflextion point.

c) It is monotically increasing so inverse will exist. Use errf table inv(B(0.66)) = 0.8

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