What are the values for h and k such that f(x) is both continuous and smooth at x=0?

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Find the values for h and k such that f(x) is both continuous and smooth at x=0

1 Answer
Apr 19, 2018

The answer h=1/4 and k=17/16

Explanation:

since f(x) is continuous and smooth:

#limf(x)xrarro^+=limf(x)xrarr0^- #
#limxrarr0^+[-(x-h)^2+K]=k-h^2#
#limxrarr0^-)[e^(0.5x)]=e^0=1#
#k-h^2=1 .........[1]#

also the function is diferentaible at x=0

#limf(x)'^+=limf(x)'^-#
#f(x)'^+=-2(x-h)#
#f(x)'^-=0.5*e^(0.5x)#

#limxrarr0^+f(0)'^+=2h#
#limxrarr0^-)f(x)'^-)=0.5#

#2h=0.5#

#h=1/4#

#k-h^2=1#
#k-(1/4)^2=1#

#k=17/16#