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Question Number 138702 by peter frank last updated on 16/Apr/21

Answered by mr W last updated on 17/Apr/21

y=sinh x+k cosh x =(√(k^2 −1)) ((1/( (√(k^2 −1))))sinh x+(k/( (√(k^2 −1)))) cosh x) =(√(k^2 −1)) (sinh α sinh x+cosh α cosh x) =(√(k^2 −1)) cosh (x+α) ≥(√(k^2 −1)) i.e. y_(min) =(√(k^2 −1)) when x+α=0, i.e. x=−α=−coth^(−1) k=−(1/2)ln ((k+1)/(k−1))=(1/2)ln ((k−1)/(k+1))

y=sinhx+kcoshx=k2−1(1k2−1sinhx+kk2−1coshx)=k2−1(sinhαsinhx+coshαcoshx)=k2−1cosh(x+α)⩾k2−1i.e.ymin=k2−1whenx+α=0,i.e.x=−α=−coth−1k=−12lnk+1k−1=12lnk−1k+1

Commented by peter frank last updated on 17/Apr/21

thank you

thankyou

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