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Question
If f(x)=4x−x2, xϵR, then write the value of f(a + 1) - (a - 1).
If f(x)=4x−x2, xϵR, then write the value of f(a + 1) - (a - 1).
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Solution
We have,
f(x)=4x−x2
Now,
f(a+1)=4(a+1)−(a+1)2=4a+4−a2−1−2a=−a2+3+2a⇒f(a+1)=−a2+2a+3
and,
f(a−1)=4(a−1)(a−1)2=4a−4−(a2+1−2a)=4a−4−a2−1+2a=6a−a2−5⇒f(a−1)=−a2+6a−5……(ii)
Subtracting equation (ii) from equation (i), we get
f(a+1)−f(a−1)=−a2+2a+3−(−a2+6a−5)=−a2+2a+3+a2−6a+5=−4a+8=4(2−a)
We have,
f(x)=4x−x2
Now,
f(a+1)=4(a+1)−(a+1)2=4a+4−a2−1−2a=−a2+3+2a⇒f(a+1)=−a2+2a+3
and,
f(a−1)=4(a−1)(a−1)2=4a−4−(a2+1−2a)=4a−4−a2−1+2a=6a−a2−5⇒f(a−1)=−a2+6a−5……(ii)
Subtracting equation (ii) from equation (i), we get
f(a+1)−f(a−1)=−a2+2a+3−(−a2+6a−5)=−a2+2a+3+a2−6a+5=−4a+8=4(2−a)
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