Optimizing a quadratic in one variable with parameterized coefficientsRecently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:0=Xtan⁡θ−g2X2sec2θ(110)2−105I want to find 0&lt;θ&lt;π2 for which I can later take the largest X value that solves this equation, i.e. optimize the implicit curve to maximize X.

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Optimizing a quadratic in one variable with parameterized coefficientsRecently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:0=Xtan⁡θ−g2X2sec2θ(110)2−105I want to find 0&amp;lt;θ&amp;lt;π2 for which I can later take the largest X value that solves this equation, i.e. optimize the implicit curve to maximize X.

Genesis Reilly · Accepted Answer

This is a similar but simple approach which gives the same result. Seeing that y=tan⁡(θ) can take any positive value, we go for maximizing x and get max(x)=1123.We have: 105=xtan⁡θ−g2x2sec2θ(110)2Let  a=105, y=tan⁡θ, b=g21(110)2.. Then,  a=xy−bx2(y2+1)Taking xy=c,;x2=c−bc2−ab (Note that since y can take any positive value, so can c)So, for ,xmax,ddc (c−bc2−ab)=0 which gives c=12bNow, xmax=c−bc2−abat,c=12b, i.e. ,xmax≈1123 by plugging in values of a and b.

Optimizing a quadratic in one variable with parameterized coefficients Recently I was doing a ph

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