Page:Calculus Made Easy.pdf/278

 (1) Min.: $$x=0$$, $$y=0$$; max.: $$x=-2$$, $$y=-4$$.

(2) $$x=a$$.

(4) $$25\sqrt{3}$$ square inches.

(5) $$\dfrac{dy}{dx} = - \dfrac{10}{x^2} + \dfrac{10}{(8 - x)^2}$$; $$x=4$$; $$y=5$$.

(6) Max. for $$x=-1$$; min. for $$x=1$$.

(7) Join the middle points of the four sides.

(8) $$r = \tfrac{2}{3} R$$, $$r = \dfrac{R}{2}$$, no max.

(9) $$r = R \sqrt{\dfrac{2}{3}}$$, $$r = \dfrac{R}{\sqrt{2}}$$, $$r = 0.8506R$$.

(10) At the rate of $$\dfrac{8}{r}$$ square feet per second.

(11) $$r=\dfrac{R \sqrt{8}}{3}$$.

(12) $$n=\sqrt{\dfrac{NR}{r}}$$.

(1) Max.: $$x=-2.19$$, $$y=24.19$$; min.: $$x=1.52$$, $$y=-1.38$$.

(2) $$\dfrac{dy}{dx}=\dfrac{b}{a}-2cx$$; $$\dfrac{d^2 y}{dx^2}=-2c$$; $$x=\dfrac{b}{2ac}$$ (a maximum).

(3) (a) One maximum and two minima. (b) One maximum. ($$x=0$$; other points unreal.)

(4) Min.: $$x=1.71$$, $$y=6.14$$.

(5) Max: $$x=-.5$$, $$y=4$$.

(6) Max.: $$x=1.414$$, $$y=1.7675$$. Min.: $$x=-1.414$$, $$y=1.7675$$.

(7) Max.: $$x=-3.565$$, $$y=2.12$$. Min.: $$x=+3.565$$, $$y=7.88$$.

(8) $$0.4N$$, $$0.6N$$.

(9) $$x=\sqrt{\dfrac{a}{c}}$$.

(10) Speed $$8.66$$ nautical miles per hour. Time taken $$115.47$$ hours. Minimum cost £$$112$$. $$12s$$.