Page:Calculus Made Easy.pdf/278

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

(2) ${\displaystyle x=a}$.

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

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

(6) Max. for ${\displaystyle x=-1}$; min. for ${\displaystyle x=1}$.

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

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

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

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

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

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

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

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

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

(4) Min.: ${\displaystyle x=1.71}$, ${\displaystyle y=6.14}$.

(5) Max: ${\displaystyle x=-.5}$, ${\displaystyle y=4}$.

(6) Max.: ${\displaystyle x=1.414}$, ${\displaystyle y=1.7675}$. Min.: ${\displaystyle x=-1.414}$, ${\displaystyle y=1.7675}$.

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

(8) ${\displaystyle 0.4N}$, ${\displaystyle 0.6N}$.

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

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