Say if the Weierstrass theorem is satisfied in the following examples and find the absolute maximum and the absolute minimum:
a) $$f(x)= \sqrt{x}$$ defined in the interval $$[-2,3.4]$$
b) $$\displaystyle f(x)=\frac{2^{\sqrt{x}-\ln x}}{4x^2+5+e^x}$$ defined in the interval $$[1,4.666666\ldots]$$
c) $$\displaystyle f(x)=3x^3+x$$ defined in the interval $$(2,4)$$
d) $$f(x)=x^2+1$$ defined in the interval $$[0,1]$$
Development:
a) We have a continuous function defined in a closed interval.
b) We have a continuous function since in the given interval we don't have to divide by zero and we do not evaluate the logarithm in points less than or equal to zero and, also, it is also a closed interval.
c) The interval is not closed.
d) We have a continuous function defined in a closed interval. The function is also strictly increasing in this interval so we will find the absolute maximum and minimum at the extremes.
We observe that $$f(0)=1$$ and $$f(1)=2$$, so at $$x=0$$ we have the absolute minimum and at $$x=1$$ we have the absolute maximum.
Solution:
a) The theorem is satisfied.
b) The theorem is satisfied.
c) The theorem is not satisfied.
d) The theorem is satisfied and we found the absolute minimum at $$x=0$$ and the absolute maximum at $$x=1$$.