Show that a function is differentiable at all pt. in R

KGPPRA

2007-10-06 8:18 am

g ( x + y ) = ( e ^ y ) g ( x ) + ( e ^ x ) g ( y )

a ) Show that g(x) is differentiable at all points in R ( real number )

b ) g'(x) is continuous in R ( real number )

回答 (1)

用戶195274

2007-10-06 9:02 am

✔ 最佳答案

(a) Note that g(0+0) = e^0 g(0) + e^0 g(0)
g(0) = 2g(0)
g(0) = 0
By first principle, for all x belongs to R,
g'(x) = lim(h→0) [g(x+h)-g(x)]/h
=lim(h→0)[e^xg(h)+e^hg(x)-g(x)]/h
=e^x lim(h→0) g(h)/h + g(x) lim(h→0)[(e^h-1)/h]
=e^x lim(h→0) g'(h)/1 + g(x) lim(h→0) (e^h/1)
=e^x g'(0) + g(x)
=g'(0)e^x + g(x)

Hence g(x) is differentiable at all points in R.

(b) For h belongs to R,
lim(h→0) g'(x+h) = lim(h→0)[g'(0)e^(x+h) + g(x+h)]
= g'(0) lim(h→0) (e^(x+h)) + lim(h→0) )[e^xg(h)+e^hg(x)]
= g'(0) e^x + e^x g(0) + e^0 g(x)
= g'(0) e^x + 0 + g(x)
= g'(0)e^x + g(x)
= g'(x)
Therefore, g' is continuous on R

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