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Solution :

We have `f(x)=x^(x)` <br> Clearly, the domain of the function is `x gt 0`. <br> Differentiating we get, `f^(')(x) = x^(x)(1+log_(e)x)` <br> Also for `x lt 1/e, f^(')(x) lt 0` and for `x gt 1//e, f^(')(x) lt 0`. <br> Thus, `x=1//e` is the point of minima. <br> `f(1//e) = (1//e)(1//e)` <br> Also, `lim_(x to 0) x^(x) =e^(lim_(x to 0)limxlogx) = e^(lim_(x to o)(1//x)/(-1//x^(2)))=e^(lim_(xto0)(-x))=e^(0)=1` <br> and `lim_(xtoinfty)x^(x)=infty` <br> From the above discussion, the graph of the function is as shown in the following figure. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/CEN_GRA_C07_E01_015_S01.png" width="80%">