Let f(x) = (e^2x)-kx for k > 0 Find local min, find y coordinate of min, find va

Question

Let f(x) = (e^2x)-kx for k > 0 Find local min, find y coordinate of min, find value of k for which this y-coordinate is largest, how do you know this value maximize k? find d^(2)y/dk^(2) to use the second derivative test

Explanation / Answer

f(x)=e^(2x)-kx for k>0 a) f '(x)=2e^(2x)-k=> f"(x)=4e^(2x) Let 2e^(2x)-k=0=> k=2e^(2x)=> 2x=ln(k/2)=> x=ln(k/2)/2 since f"(x)=4(k/2)=k/2>0 x=ln(k/2)/2 gives a min. of f. b) y(min.)= (k/2)-k[ln(k/2)/2]= (k/2)[1-ln(k/2)] c) y'(min)=1/2-ln(k/2)/2-1/2=> y'(min)=-ln(k/2)/2 y'(min.)=0=> k/2=1=> k=2 d) y"(min.)=-1/(2k) k=2 gives the y(min.) the largest.

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