<p><strong>Let <math><mrow><mi>c</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>k</mi><mo>∈</mo><mi>R</mi><mo>.</mo><mtext> </mtext><mtext> </mtext><mi>I</mi><mi>f</mi><mtext> </mtext><mtext> </mtext><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow><mi>x</mi><mo>+</mo><mn>2</mn><mi>k</mi></mrow></math> and <math><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>y</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>x</mi><mi>y</mi><mo>,</mo></mrow></math> for all x, y <math><mrow><mo>∈</mo></mrow></math> R, then the value of <math><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mn>2</mn><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow></math> is equal to………….</strong></p>

f (x) = (c + 1)x2 + (1 – c2)x + 2kf' (x)= ( (c+1))2x+ (1−c2)............. (i)f' (x)=limy→0If (x+y)−f (x)x+y−x=limy→0f (y)−xyyf' (x)=limy→0f (y)y−xx+f' (x)=limy→0f (y)−f (0)y−0as f (c)=0

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Maths Class 12th Binary Operation Maths NCERT Exemplar Solutions Class 12th Chapter Twelve

Let $c, k \in R . I f f (x) = (c + 1) x^{2} + (1 - c^{2}) x + 2 k$ and $f (x + y) = f (x) + f (y) - x y,$ for all x, y $\in$ R, then the value of $| 2 (f (1) + f (2) + f (3) + . . . . + f (20)) |$ is equal to………….

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alok kumar singh | Contributor-Level 10

2 months ago

f (x) = (c + 1)x² + (1 – c²)x + 2k

$f' (x) = ((c + 1)) 2 x + (1 - c^{2}) . . . . . . . . . . . . . (i)$

$f' (x) = \underset{y \to 0}{l i m} I \frac{f (x + y) - f (x)}{x + y - x}$

$= \underset{y \to 0}{l i m} \frac{f (y) - x y}{y}$

$f' (x) = \underset{y \to 0}{l i m} \frac{f (y)}{y} - x$

$x + f' (x) = \underset{y \to 0}{l i m} \frac{f (y) - f (0)}{y - 0} a s f (c) = 0$

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Let $c, k \in R . I f f (x) = (c + 1) x^{2} + (1 - c^{2}) x + 2 k$ and $f (x + y) = f (x) + f (y) - x y,$ for all x, y $\in$ R, then the value of $| 2 (f (1) + f (2) + f (3) + . . . . + f (20)) |$ is equal to………….

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Let c, k∈R. If f(x)=(c+1)x2+(1−c2)x+2k and f(x+y)=f(x)+f(y)−xy, for all x, y ∈ R, then the value of |2(f(1)+f(2)+f(3)+....+f(20))| is equal to………….

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Let $c, k \in R . I f f (x) = (c + 1) x^{2} + (1 - c^{2}) x + 2 k$ and $f (x + y) = f (x) + f (y) - x y,$ for all x, y $\in$ R, then the value of $| 2 (f (1) + f (2) + f (3) + . . . . + f (20)) |$ is equal to………….