What is the minimum value of a sec x + b csc | vedclass

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(C) Let $y = a \sec x + b \csc x$.To find the minimum value,we differentiate with respect to $x$:$\frac{dy}{dx} = a \sec x 	an x - b \csc x \cot x$.S

Vedclass · Accepted Answer

$(a^{2/3} + b^{2/3})^{3/2}$

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(C) Let $y = a \sec x + b \csc x$.To find the minimum value,we differentiate with respect to $x$:$\frac{dy}{dx} = a \sec x 	an x - b \csc x \cot x$.Setting $\frac{dy}{dx} = 0$ for critical points:$a \sec x 	an x = b \csc x \cot x$.Converting to sine and cosine:$\frac{a \sin x}{\cos^2 x} = \frac{b \cos x}{\sin^2 x} \implies a \sin^3 x = b \cos^3 x \implies 	an^3 x = \frac{b}{a}$.Thus,$	an x = (b/a)^{1/3}$.Using the identity $\sec^2 x = 1 + 	an^2 x$ and $\csc^2 x = 1 + \cot^2 x$:$\sec x = \sqrt{1 + (b/a)^{2/3}} = \frac{\sqrt{a^{2/3} + b^{2/3}}}{a^{1/3}}$ and $\csc x = \sqrt{1 + (a/b)^{2/3}} = \frac{\sqrt{a^{2/3} + b^{2/3}}}{b^{1/3}}$.Substituting these into the expression for $y$:$y = a \left( \frac{\sqrt{a^{2/3} + b^{2/3}}}{a^{1/3}} ight) + b \left( \frac{\sqrt{a^{2/3} + b^{2/3}}}{b^{1/3}} ight)$.$y = a^{2/3} \sqrt{a^{2/3} + b^{2/3}} + b^{2/3} \sqrt{a^{2/3} + b^{2/3}}$.$y = (a^{2/3} + b^{2/3}) \sqrt{a^{2/3} + b^{2/3}} = (a^{2/3} + b^{2/3})^{3/2}$.

What is the minimum value of $a \sec x + b \csc x$ for $0 < a < b$ and $0 < x < \pi/2$?

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