If f(x)=1/(4x^2  + 2x + 1);x∈R, then find the maximum value of f(x).

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f(𝑥)=1/(4𝑥^2 + 2𝑥 + 1) Finding f’(𝒙) f’(𝑥)= ((1)^′ " " (4𝑥^2 + 2𝑥 + 1)" − " (〖4𝑥^2 + 2𝑥 + 1)〗^′ (1))/((〖4𝑥^2 + 2𝑥 + 1)〗^2 ) f’(𝑥)= (0 (4𝑥^2 + 2𝑥 + 1)" − " (8𝑥 + 2)(1))/((〖4𝑥^2 + 2𝑥 + 1)〗^2 ) f’(𝑥)= ("−" (8𝑥 + 2) )/((〖4𝑥^2 + 2𝑥 + 1)〗^2 ) Putting f’(𝒙)=𝟎 ("−" (8𝑥 + 2) )/((〖4𝑥^2 + 2𝑥 + 1)〗^2 ) = 0 -(8x + 2) = 0 8x + 2 = 0 8x = -2 −(8x + 2) = 0 8x + 2 = 0 8x = −2 x = (−2)/8 x = (−𝟏)/𝟒 Finding f’’(𝒙) f’(𝑥)=("−" (8𝑥 + 2) )/((〖4𝑥^2 + 2𝑥 + 1)〗^2 ) " " Differentiating again w.r.t x f’’(x) =−((8𝑥 + 2)^′ (〖4𝑥^2 + 2𝑥 + 1)〗^2−((〖4𝑥^2+2𝑥+1)〗^2 )^′ (8𝑥 + 2))/(((〖4𝑥^2 + 2𝑥 + 1)〗^2 )^2 ) f’’(x) =−(8(〖4𝑥^2 + 2𝑥 + 1)〗^2 − 2(4𝑥^2 + 2𝑥 + 1)(8𝑥 + 2)(8𝑥 + 2))/(4𝑥^2 + 2𝑥 + 1)^4 f’’(x) =−(8(〖4𝑥^2 + 2𝑥 + 1)〗^2 − 2(4𝑥^2 + 2𝑥 + 1)(8𝑥 + 2)(8𝑥 + 2))/(4𝑥^2 + 2𝑥 + 1)^4 f’’(x) =−(8(〖4𝑥^2 + 2𝑥 + 1)〗^2 − 2(4𝑥^2 + 2𝑥 + 1) (8𝑥 + 2)^2)/(4𝑥^2 + 2𝑥 + 1)^4 f’’ (−𝟏/𝟒) = −(8(〖4(−1/4)^2+ 2(−1/4) + 1)〗^2 − 2(4(−1/4)^2+ 2(−1/4)+ 1) (8(−1/4)+ 2)^2)/(4(−1/4)^2+ 2(−1/4)+ 1)^4 f’’ (−𝟏/𝟒) = −(8(〖4(−1/4)^2+ 2(−1/4) + 1)〗^2 − 2(4(−1/4)^2+ 2(−1/4)+ 1) (−2 + 2)^2)/(4(−1/4)^2+ 2(−1/4)+ 1)^4 f’’ (−𝟏/𝟒) = −(8(3/4)^2−0)/(3/4)^4 = −8/(3/4)^2 f’’ (−𝟏/𝟒) < 0 Since f’’ (−𝟏/𝟒) < 0 , 𝑥 = −𝟏/𝟒 is point of local maxima Putting 𝑥 = −𝟏/𝟒 , we can calculate maximum value f(𝑥) =1/(4𝑥^2+2𝑥+1) f(−𝟏/𝟒)=1/(4(−1/4)^2+ 2(−1/4)+ 1) =1/(4(1/16)+ 2(−1/4)+ 1) =1/(1/4 − 2/4+ 1) = 4/(1 −2+ 4) = 𝟒/𝟑

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo