Find-maximum-value-of-function-x-2x-16-x-35-x-

Question Number 151685 by iloveisrael last updated on 22/Aug/21

Find maximum value of function α(x)= (√(2x)) +(√(16−x)) +(√(35+x)) .

$$\:\:{Find}\:{maximum}\:{value}\:{of}\:{function} \\ $$$$\:\:\alpha\left({x}\right)=\:\sqrt{\mathrm{2}{x}}\:+\sqrt{\mathrm{16}−{x}}\:+\sqrt{\mathrm{35}+{x}}\:. \\ $$

Answered by mr W last updated on 22/Aug/21

x≥0 x≤16 f′(x)=(1/( (√(2x))))−(1/( 2(√(16−x))))+(1/( 2(√(35+x))))=0 (2/( (√(2x))))+(1/( (√(35+x))))=(1/( (√(16−x)))) (2/( x))+(1/( 35+x))+((2(√2))/( (√(x(35+x)))))=(1/( 16−x)) ((2(√2))/( (√(x(35+x)))))=((x(35+x)−x(16−x)−2(35+x)(16−x))/( (16−x)(x)(35+x))) ((2(√2))/( (√(x(35+x)))))=((4x^2 +57x−1120)/( (16−x)(x)(35+x))) 8=(((4x^2 +57x−1120)^2 )/( (16−x)^2 (x)(35+x))) 8x^4 +432x^3 +1201x^2 −199360x+1254400=0 ⇒x≈12.6106 f(x)_(max) ≈13.7631

$${x}\geqslant\mathrm{0} \\ $$$${x}\leqslant\mathrm{16} \\ $$$${f}'\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}}}−\frac{\mathrm{1}}{\:\mathrm{2}\sqrt{\mathrm{16}−{x}}}+\frac{\mathrm{1}}{\:\mathrm{2}\sqrt{\mathrm{35}+{x}}}=\mathrm{0} \\ $$$$\frac{\mathrm{2}}{\:\sqrt{\mathrm{2}{x}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{35}+{x}}}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{16}−{x}}} \\ $$$$\frac{\mathrm{2}}{\:{x}}+\frac{\mathrm{1}}{\:\mathrm{35}+{x}}+\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\:\sqrt{{x}\left(\mathrm{35}+{x}\right)}}=\frac{\mathrm{1}}{\:\mathrm{16}−{x}} \\ $$$$\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\:\sqrt{{x}\left(\mathrm{35}+{x}\right)}}=\frac{{x}\left(\mathrm{35}+{x}\right)−{x}\left(\mathrm{16}−{x}\right)−\mathrm{2}\left(\mathrm{35}+{x}\right)\left(\mathrm{16}−{x}\right)}{\:\left(\mathrm{16}−{x}\right)\left({x}\right)\left(\mathrm{35}+{x}\right)} \\ $$$$\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\:\sqrt{{x}\left(\mathrm{35}+{x}\right)}}=\frac{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{57}{x}−\mathrm{1120}}{\:\left(\mathrm{16}−{x}\right)\left({x}\right)\left(\mathrm{35}+{x}\right)} \\ $$$$\mathrm{8}=\frac{\left(\mathrm{4}{x}^{\mathrm{2}} +\mathrm{57}{x}−\mathrm{1120}\right)^{\mathrm{2}} }{\:\left(\mathrm{16}−{x}\right)^{\mathrm{2}} \left({x}\right)\left(\mathrm{35}+{x}\right)} \\ $$$$\mathrm{8}{x}^{\mathrm{4}} +\mathrm{432}{x}^{\mathrm{3}} +\mathrm{1201}{x}^{\mathrm{2}} −\mathrm{199360}{x}+\mathrm{1254400}=\mathrm{0} \\ $$$$\Rightarrow{x}\approx\mathrm{12}.\mathrm{6106} \\ $$$${f}\left({x}\right)_{{max}} \approx\mathrm{13}.\mathrm{7631} \\ $$

Commented by mr W last updated on 22/Aug/21