# Calculating error bounds when $\max f^{(2)}(x)$ is undefined

The error bound for Trapezoidal rule is: $${\left|E_{T}\right| \leq k \frac{(b-a)^{3}}{12 n^{2}}}$$

I am trying to calculate the error for $$x^x$$ in the interval $$[0,1]$$ and let $$n = 500$$. The problem is I am not able to compute $$k = \max (f^{(2)}(x))$$ because $$\max (f^{(2)}(x))$$is undefined. How do I go about calculating the error bound?

The second derivative of $$f(x)=x^x$$ is $$f^{(2)}(x)=x^x((\ln(x)+1)^2+1/x)$$ which is continuous, positive and convex in $$(0,1]$$ . Hence for $$0 $$\max_{[a,1]}|f^{(2)}(x)|=\max(f(a),f(1))=\max(f(a),2).$$ Unfortunately $$\lim_{a\to 0^+}f^{(2)}(a)=+\infty$$ therefore you should apply the Trapezoidal rule to a smaller interval $$[a,1]$$ with $$0.
Note that $$x^x$$ is integrable in $$[0,1]$$ and for $$0 we can easily estimate the error on the missing part $$[0,a]$$: $$0<\int_0^a x^x \,dx< a$$ because $$0 in $$[0,1]$$.
P.S. The following identity (see Sophomore's dream) $$\int_0^1x^x\ dx=\sum_{n=1}^\infty(-1)^{n+1}n^{-n}\approx 0.7834305107.$$ allows you to approximate the given integral quite efficiently.
• Im so sorry. There was a typo in the question. I meant $[0,1]$ Nov 9 '19 at 8:27