If f(x)=4x-x^2, x , then write the value of f(a + 1) - f(a - 1). - Vidyadip

Question

If $\text{f(x)}=4\text{x}-\text{x}^2,\text{ x}\in\text{R},$ then write the value of $f(a + 1) - f(a - 1)$.

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Answer

We have, $f(x)=4 x-x^2$ Now, $f(a+1)=4(a+1)-(a+1)^2=4 a+4-a^2-1-2 a=-a^2+3+2 a \Rightarrow f(a+1)=-a^2+2 a+$ $3 \ldots(\mathrm{i})$ and, $f(a-1)=4(a-1)-(a-1)^2=4 a-4-\left(a^2+1-2 a\right)=4 a-4-a^2-1+2 a=6 a-a-5 f(a-1)=-a^2+6 a-5 \ldots$. (ii) Subtracting equation (ii) from equation (i), we get $f(a+1)-f(a-1)=-a^2+2 a+3-\left(-a^2+6 a-5\right)=-a^2+2 a+3+$ $a^2-6 a+5=-4 a+8=4(2-a)$

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