Let <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>|</mo></math> and <math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>,</mo><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0,4</mn><mo>]</mo></math> . Then <math><msubsup><mrow><mrow><mo>∫</mo></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msubsup><mo>?</mo><mo>(</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>x</mi></math> is equal to:

∫03? g (x)-f (x)=∫03? ||x-2|-2|dx-∫03? |x-2|dx=12×2×2+1+12×1×1-12×2×2+12×1×1=2+1+12-2+12=1

Let&nbsp;<math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>|</mo></math>&nbsp;and&nbsp;<math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>,</mo><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0,4</mn><mo>]</mo></math>&nbsp;. Then&nbsp;<math><msubsup><mrow><mrow><mo>∫</mo></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msubsup><mo>?</mo><mo>(</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>x</mi></math>&nbsp;is equal to:

Ask & Answer Question

Option 1 -

$\frac{1}{2}$

Option 2 -

$\frac{3}{2}$

Option 3 -

Option 4 -

3 Views | Posted 2 months ago

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 3

Detailed Solution:

$\int_{0}^{3} ? g (x) - f (x) = \int_{0}^{3} ? | | x - 2 | - 2 | d x - \int_{0}^{3} ? | x - 2 | d x$

$= (\frac{1}{2} \times 2 \times 2 + 1 + \frac{1}{2} \times 1 \times 1) - (\frac{1}{2} \times 2 \times 2 + \frac{1}{2} \times 1 \times 1)$

$= (2 + 1 + \frac{1}{2}) - (2 + \frac{1}{2}) = 1$