Ask & Answer Question

5f(x) + 4f $(\frac{1}{x})$ = x² – 4 and y = 9f(x) × x². If y is strictly increasing function, find interval of x.

Option 1 -

$(- \infty, \frac{- 1}{\sqrt[]{5}}] \cup (\frac{- 1}{\sqrt[]{5}}, 0)$

Option 2 -

$(\frac{- 1}{\sqrt[]{5}}, 0) \cup (0, \frac{1}{\sqrt[]{5}})$

Option 3 -

$(0, \frac{1}{\sqrt[]{5}}) \cup (\frac{1}{\sqrt[]{5}}, \infty)$

Option 4 -

$(- \sqrt[]{\frac{2}{5}}, 0) \cup (\sqrt[]{\frac{2}{5}}, \infty)$

2 Views | Posted a week ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a week ago

Correct Option - 4

Detailed Solution:

5f(x) + 4f $(\frac{1}{x})$ = x² – 4 .(1)

Replace x by $\frac{1}{x}$

5f $(\frac{1}{x})$ + 4f(x) = $\frac{1}{x^{2}}$ – 4 .(2)

5 × equation (1) – 4 × equation (2)

$9 f (x) = 5 x^{2 -} \frac{4}{x^{2}} - 4$

$y = 9 f (x) \cdot x^{2} = \frac{5 x^{4} - 4 - 4 x^{2}}{x^{2}} x^{2}$

y = 5x⁴ – 4 – 4x²

y = 20x³ – 8x > 0

4x(5x² – 2) > 0

$x \in (- \sqrt[]{\frac{2}{5}}, 0) \cup (\sqrt[]{\frac{2}{5}}, \infty)$

5f(x) + 4f <math> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math>  = x2 – 4           .(1)Replace x by  <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </math> 5f  <math> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math>  + 4f(x) = <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </math>  – 4   .(2)5 × equation (1) – 4 × equation (2) <math> <mrow> <mn>9</mn> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>5</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mo>−</mo> </mrow> </msup> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>−</mo> <mn>4</mn> </mrow> </math>             <math> <mrow> <mi>y</mi> <mo>=</mo> <mn>9</mn> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mo>−</mo> <mn>4</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>            y = 5x4 – 4 – 4x2y = 20x3 – 8x > 04x(5x2 – 2) > 0<picture><img src="https://images.shiksha.com/mediadata/images/1758801580phpa1HNFo.jpeg" alt="" width="316" height="133" loading="lazy"></picture>    <math> <mrow> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mroot> <mrow> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> <mo>,</mo> <mtext> </mtext> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>∪</mo> <mrow> <mo>(</mo> <mrow> <mroot> <mrow> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> <mo>,</mo> <mtext> </mtext> <mo>∞</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math>

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
688k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

5f(x) + 4f $(\frac{1}{x})$ = x² – 4 and y = 9f(x) × x². If y is strictly increasing function, find interval of x.

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

5f(x) + 4f ( 1 x ) = x2 – 4 and y = 9f(x) × x2. If y is strictly increasing function, find interval of x.

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

5f(x) + 4f $(\frac{1}{x})$ = x² – 4 and y = 9f(x) × x². If y is strictly increasing function, find interval of x.