Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
BITSAT 2013
Exam
BITSAT
Year
2013
Subject
Mathematics
Chapter
Application of Derivatives

BITSAT 2013 · Mathematics · Application of Derivatives

If a^ 2 x^ 4 +b^ 2 y^ 4 =c^ 4 , then the maximum value of xy is

If a2x4+b2y4=c4a^{2} x^{4}+b^{2} y^{4}=c^{4}, then the maximum value of xy is

Options

  1. Acab\frac{\mathrm{c}}{\sqrt{\mathrm{ab}}}
  2. Bc22ab\frac{c^{2}}{2 \sqrt{a b}}
  3. Cc2ab\frac{\mathrm{c}}{2 \sqrt{\mathrm{ab}}}
  4. Dc22ab\frac{c^{2}}{\sqrt{2 a b}}

Correct answer

D. c22ab\frac{c^{2}}{\sqrt{2 a b}}

Step-by-step solution

If the sum of two positive quantities is a constant, then their product is maximum, when they are equal. ∴\mathrma2\mathrmx4⋅\mathrmb2\mathrmy2\therefore \mathrma^{2} \mathrmx^{4} \cdot \mathrmb^{2} \mathrmy^{2} is maximum when a^{2} x^{4}=b^{2} y^{4}=12(a2x4=12(a2x4+b2y4)=c42∴\frac{1}{2}\left(a^{2} x^{4}=\frac{1}{2}\left(a^{2} x^{4}+b^{2} y^{4}\right)=\frac{c^{4}}{2}\right.\therefore maximum value of a2x4⋅b2y4=c42⋅c42=c84a^{2} x^{4} \cdot b^{2} y^{4}=\frac{c^{4}}{2} \cdot \frac{c^{4}}{2}=\frac{c^{8}}{4} Maximum value of xy=(c84a2b2)1/4=c22abx y=\left(\frac{c^{8}}{4 a^{2} b^{2}}\right)^{1 / 4}=\frac{c^{2}}{\sqrt{2 a b}}
Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

The maximum area of rectangle inscribed in a circle of diameter R is BITSAT 2024Consider the function f(x)=(|x-1|)/(x^2) , then f(x) is BITSAT 2024The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of… BITSAT 2024If the angle made by the tangent at the point (x_0, y_0) on the curve x=12(t+ t t)… BITSAT 2024The altitude of a cone is 20 cm and its semivertical angle is 30^ ^ ∘ . If the… BITSAT 2024The point of inflexion for the curve y=(x-a)^n , where n is odd integer and n ≥ 3 is BITSAT 2024The population p(t) at time t of a certain mouse species satisfies the differential… BITSAT 2024The function f(x)= (1+x)-(2 x)/(2+x) is increasing on BITSAT 2023If f(x)= x, g(x)= 2 x, h(x)= 3 x and I(x)= x , then which of the following option is… BITSAT 2023The interval in which the function f(x)=(4 x^2+1)/(x) is decreasing is : BITSAT 2023The curve given by x+y=e^ x y has a tangent parallel to the Y-axis at the point BITSAT 2023The function f(x)= ^ -1 ( x+ x) is an increasing function in BITSAT 2023A running track of 440 ft is to be laid out enclosing a football field, the shape of… BITSAT 2022A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the… BITSAT 2022The points at which the tangent passes through the origin for the curve y=4 x^ 3 -2 x^ 5… BITSAT 2021If the radius of a sphere is measured as 9 cm with an error of 0.03 cm , then find the… BITSAT 2021 Full Application of Derivatives list Application of Derivatives JEE Maths PYQsJEE Mathematics PYQs All BITSAT PYQs