Question: Smoothness (10 points). Prove that if a differentiable function f:RdRis-smooth, thenfor all v,winRd,we have:f(v)f(w)+(:gradf(w),v-w:)+2||v-w||22Hint: For any v,winRd, define a function g:RRbyg(t)=f(w+t(v-w)). Apply thefundamental theorem of

Smoothness (10 points). Prove that if a differentiable function f:RdRis-smooth, thenfor all v,winRd,we have:f(v)f(w)+(:gradf(w),v-w:)+2||v-w||22Hint: For any v,winRd, define a function g:RRbyg(t)=f(w+t(v-w)). Apply thefundamental theorem of calculus tog(1)-g(0)to obtain:g(1)-g(0)=01g'(t)dtand proceed from there.

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