The minimum values of a for which the equation $\frac{4}{s i n x} + \frac{1}{1 - s i n x} = α$ has at least one solution in $(0, \frac{π}{2}) i s_____.$

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alok kumar singh | Contributor-Level 10

2 months ago

$\frac{4}{s i n x} + \frac{1}{1 - s i n x} = a \forall x \in (0, \frac{π}{2})$

Let sin x = t, t $\in$ (0, 1)

$g (t) = \frac{4}{t} + \frac{1}{1 - t}$

$g' (t) = 0 \Rightarrow t = \frac{2}{3}$

$g'' (\frac{2}{3}) > 0$

$∴ g {(t)}_{m i n i m u m} = \frac{4}{2 / 3} + \frac{1}{1 - 2 / 3} = 9$

Minimum value of a for which solution exist = 9

<!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816619397"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>a</mi> <mo>∀</mo> <mi>x</mi> <mo>∈</mo> <mrow> <mo> (</mo> <mrow> <mn>0</mn> <mo>, </mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math>   Let sin x = t, t <math> <mrow> <mo>∈</mo> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816619398"> </o:OLEObject></xml><! [endif]-> (0, 1) <math> <mrow> <mi>g</mi> <mrow> <mo> (</mo> <mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mi>g</mi> <mo>'</mo> <mrow> <mo> (</mo> <mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>⇒</mo> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mi>g</mi> <mo>'</mo> <mo>'</mo> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>></mo> <mn>0</mn> </mrow> </math> <math> <mrow> <mo>∴</mo> <mi>g</mi> <msub> <mrow> <mrow> <mo> (</mo> <mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mi>i</mi> <mi>m</mi> <mi>u</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </mfrac> <mo>=</mo> <mn>9</mn> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1816619399"> </o:OLEObject></xml><! [endif]-> Minimum value of a for which solution exist = 9

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The minimum values of a for which the equation $\frac{4}{s i n x} + \frac{1}{1 - s i n x} = α$ has at least one solution in $(0, \frac{π}{2}) i s_____.$

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The minimum values of a for which the equation 4 s i n x + 1 1 − s i n x = α has at least one solution in ( 0 , π 2 ) i s _ _ _ _ _ .

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The minimum values of a for which the equation $\frac{4}{s i n x} + \frac{1}{1 - s i n x} = α$ has at least one solution in $(0, \frac{π}{2}) i s_____.$