Question-81771

Question Number 81771 by Power last updated on 15/Feb/20

Commented by Power last updated on 15/Feb/20

$$\mathrm{solution} \\ $$

Commented by Power last updated on 15/Feb/20

What value of the parameter ↑ will have 2 roots ?

$$\mathrm{What}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{parameter}\:\:\uparrow\:\:\mathrm{will}\:\mathrm{have}\:\mathrm{2}\:\mathrm{roots}\:? \\ $$

Commented by mr W last updated on 15/Feb/20

two roots when −4≤a<2 one root when a=2 no root when a<−4 or a>2

$${two}\:{roots}\:{when}\:−\mathrm{4}\leqslant{a}<\mathrm{2} \\ $$$${one}\:{root}\:{when}\:{a}=\mathrm{2} \\ $$$${no}\:{root}\:{when}\:{a}<−\mathrm{4}\:{or}\:{a}>\mathrm{2} \\ $$

Answered by mr W last updated on 15/Feb/20

let f(x)=(√(4−∣2x∣)) let g(x)=x^2 +a both functions are even, i.e. f(−x)=f(x), g(−x)=g(x) we only need to look at x≥0. f(x)=(√(4−2x)) : 0≤x≤2 f(0)=2=maximum f(2)=0=minimum g(x)=x^2 +a : this is a parabola opening upwards, its lowest point is (0,a). such that the eqn. has two roots, the curve g(x) must intersect the curve g(x) at two points. we have following cases: case 1: a>2, g(x) is too high and can′t intersect the curve f(x) ⇒no root. case 2: a=2, the lowest point of g(x) coincides with the topmost point of f(x), one intersection point ⇒one root. case 3: −4≤a<2, g(x) intersects with f(x) at two points ⇒two roots. case 4: a=−4, y=x^2 −4, y=0 ⇒x=±2. f(x) intersects with g(x) at (−2,0) and (2,0)⇒two roots. case 5: a<−4, g(x) is too low and can′t intersect with f(x)⇒no root.

$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{4}−\mid\mathrm{2}{x}\mid} \\ $$$${let}\:{g}\left({x}\right)={x}^{\mathrm{2}} +{a} \\ $$$${both}\:{functions}\:{are}\:{even},\:{i}.{e}.\: \\ $$$${f}\left(−{x}\right)={f}\left({x}\right),\:{g}\left(−{x}\right)={g}\left({x}\right) \\ $$$${we}\:{only}\:{need}\:{to}\:{look}\:{at}\:{x}\geqslant\mathrm{0}. \\ $$$${f}\left({x}\right)=\sqrt{\mathrm{4}−\mathrm{2}{x}}\:: \\ $$$$\mathrm{0}\leqslant{x}\leqslant\mathrm{2} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{2}={maximum} \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{0}={minimum} \\ $$$${g}\left({x}\right)={x}^{\mathrm{2}} +{a}\:: \\ $$$${this}\:{is}\:{a}\:{parabola}\:{opening}\:{upwards}, \\ $$$${its}\:{lowest}\:{point}\:{is}\:\left(\mathrm{0},{a}\right). \\ $$$$ \\ $$$${such}\:{that}\:{the}\:{eqn}.\:{has}\:{two}\:{roots},\:{the} \\ $$$${curve}\:{g}\left({x}\right)\:{must}\:{intersect}\:{the}\:{curve}\:{g}\left({x}\right) \\ $$$${at}\:{two}\:{points}. \\ $$$${we}\:{have}\:{following}\:{cases}: \\ $$$$ \\ $$$${case}\:\mathrm{1}:\:{a}>\mathrm{2},\:{g}\left({x}\right)\:{is}\:{too}\:{high}\:{and}\:{can}'{t} \\ $$$${intersect}\:{the}\:{curve}\:{f}\left({x}\right)\:\Rightarrow{no}\:{root}. \\ $$$$ \\ $$$${case}\:\mathrm{2}:\:{a}=\mathrm{2},\:{the}\:{lowest}\:{point}\:{of}\:{g}\left({x}\right)\: \\ $$$${coincides}\:{with}\:{the}\:{topmost}\:{point}\:{of} \\ $$$${f}\left({x}\right),\:{one}\:{intersection}\:{point}\:\Rightarrow{one} \\ $$$${root}. \\ $$$$ \\ $$$${case}\:\mathrm{3}:\:−\mathrm{4}\leqslant{a}<\mathrm{2},\:{g}\left({x}\right)\:{intersects}\:{with} \\ $$$${f}\left({x}\right)\:{at}\:{two}\:{points}\:\Rightarrow{two}\:{roots}. \\ $$$$ \\ $$$${case}\:\mathrm{4}:\:{a}=−\mathrm{4},\:{y}={x}^{\mathrm{2}} −\mathrm{4},\:{y}=\mathrm{0}\:\Rightarrow{x}=\pm\mathrm{2}. \\ $$$${f}\left({x}\right)\:{intersects}\:{with}\:{g}\left({x}\right)\:{at}\:\left(−\mathrm{2},\mathrm{0}\right)\:{and} \\ $$$$\left(\mathrm{2},\mathrm{0}\right)\Rightarrow{two}\:{roots}. \\ $$$$ \\ $$$${case}\:\mathrm{5}:\:{a}<−\mathrm{4},\:{g}\left({x}\right)\:{is}\:{too}\:{low}\:{and}\:{can}'{t} \\ $$$${intersect}\:{with}\:{f}\left({x}\right)\Rightarrow{no}\:{root}. \\ $$

Commented by mr W last updated on 15/Feb/20