Suppose f; g are differentiable functions from R to Rn.(a) Show thathf .t /; g.t /i0 D

Chapter 6, Problem 26

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Suppose f; g are differentiable functions from R to Rn.(a) Show thathf .t /; g.t /i0 D hf 0.t /; g.t /iChf .t /; g0.t /i:(b) Suppose c>0 and kf .t /k D c for every t 2 R. Show thathf 0.t /; f .t /i D 0 for every t 2 R.(c) Interpret the result in part (b) geometrically in terms of the tangentvector to a curve lying on a sphere in Rn centered at the origin.[For the exercise above, a function f W R ! Rn is called differentiableif there exist differentiable functions f1;:::;fn from R to R such thatf .t / Df1.t /; : : : ; fn.t /for each t 2 R. Furthermore, for each t 2 R,the derivative f 0.t / 2 Rn is defined by f 0.t / Df10.t /; : : : ; fn0.t /.]