Math, asked by ExchangeMyStack, 9 months ago

If
f(x) =  \:  \frac{1}{2} (2x {}^{3}  -  {x}^{2}  + x)
Then find,
f(x + 1) - f(x)

Answers

Answered by Rohit18Bhadauria
4

Given:

\bf{f(x)=\dfrac{1}{2}(2x^{3}-x^{2}+x)}

To Find:

  • Value of f(x+1)-f(x)

​Solution:

We know that,

  • (a+b)³ =a³+b³+3ab(a+b)
  • (a+b)² =a²+b²+2ab

On substituting x with x+1 in the expression of f(x), we get

\sf{f(x+1)=\dfrac{1}{2}(2(x+1)^{3}-(x+1)^{2}+x+1)}

Now,

\sf{f(x+1)-f(x)=\dfrac{1}{2}(2(x+1)^{3}-(x+1)^{2}+x+1)-\dfrac{1}{2}(2x^{3}-x^{2}+x)}

\sf{\implies\dfrac{1}{2}(2(x^{3} +1+3x(x+1))-(x^{2} +1+2x)+x+1)-\dfrac{1}{2}(2x^{3}-x^{2}+x)}

\sf{\implies\dfrac{1}{2}(2(x^{3} +1+3x^{2}+3x)-x^{2}-1-2x+x+1)-\dfrac{1}{2}(2x^{3}-x^{2}+x)}

\sf{\implies\dfrac{1}{2}(2x^{3} +2+6x^{2}+6x-x^{2}-1-2x+x+1)-\dfrac{1}{2}(2x^{3}-x^{2}+x)}

\sf{\implies\dfrac{1}{2}(2x^{3} +2+6x^{2}+6x-x^{2}-1-2x+x+1-2x^{3}+x^{2}-x)}

\sf{\implies\dfrac{1}{2}(6x^{2}+4x+2)}

\sf\pink{\implies3x^{2}+2x+1}

Hence, the value of f(x+1)-f(x) is 3x²+2x+1.

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