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Method of Differentiation: Let Polynomial Function Second Degree

JEE Maths question with a full step-by-step solution.

Question
Let f(x)f(x) be a polynomial function of second degree. If f(1)=f(−1)f(1)=f(-1) and a,b,ca,b,c are in A.P., then f′(a),f′(b),f′(c)f'(a),f'(b),f'(c) are in
AAPcorrect
BGP
CHP
DAGP
Solution
Step 1: Let f(x)=αx2+βx+γf(x)=\alpha x^{2}+\beta x+\gamma. Step 2: Apply the condition f(1)=f(−1)f(1)=f(-1):
α+β+γ=α−β+γ  ⇒  2β=0  ⇒  β=0\alpha+\beta+\gamma=\alpha-\beta+\gamma\;\Rightarrow\;2\beta=0\;\Rightarrow\;\beta=0
 So f(x)=αx2+γf(x)=\alpha x^{2}+\gamma and f′(x)=2αxf'(x)=2\alpha x. Step 3: Since a,b,ca,b,c are in A.P., we have a+c=2ba+c=2b.
f′(a)+f′(c)=2αa+2αc=2α(a+c)=2α(2b)=2(2αb)=2f′(b)f'(a)+f'(c)=2\alpha a+2\alpha c=2\alpha(a+c)=2\alpha(2b)=2(2\alpha b)=2f'(b)
 Therefore f′(a),f′(b),f′(c)f'(a),f'(b),f'(c) are in A.P. Correct answer: (1)
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