91Ó°ÊÓ

A box of max volume with top open is to be made out of a square tin sheet of sides \(6 \mathrm{ft}\) length by cutting out small equal squares from four corners of the sheet. Find the height of the box.

Find all the values of the parameter \(a\) for which the point of minimum of the function \(f(x)=1+a^{2} x-x^{3}\) satisfies the inequality \(\frac{x^{2}+x+2}{x^{2}+5 x+6}<0\).

Min. area of the triangle formed by any tangent to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with the co-ordinate axes is (a) \(\frac{a^{2}+b^{2}}{2}\) (b) \(\frac{(a+b)^{2}}{2}\) (c) \(a b\) (d) \(\frac{(a-b)^{2}}{2}\)

Rectangle of max. area that can be inscribed in an equilateral \(\Delta\) of side a will have area = (a) \(\frac{a^{2} \sqrt{3}}{2}\) (b) \(\frac{a^{2} \sqrt{3}}{4}\) (c) \(\frac{a^{2} \sqrt{3}}{8}\) (d) None

Find the min values of \(f(x)=2^{x}+3^{x}+5^{x}+\frac{1}{2^{x}}+\frac{1}{3^{x}}+\frac{1}{5^{x}}, x>0\)