Ex 6.2,11 - Chapter 6 Class 12 Application of Derivatives

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Ex 6.2, 11 Prove that the function f given by f (𝑥) = 𝑥^2 – 𝑥 + 1 is neither strictly increasing nor strictly decreasing on (– 1, 1). Let f(𝑥) = 𝑥2 – 𝑥 + 1 𝑥 ∈ (−1 , 1) Finding f’(𝒙) f’(𝑥) = 2𝑥 – 1 Putting f’(𝒙) = 0 2𝑥 – 1 = 0 2𝑥 = 1 𝑥 = 1/2 Since 𝑥 ∈ (−1 , 1), So, our number line looks like The point 𝑥 = 1/2 divide the intervals (−1 , 1) into two disjoint intervals. i.e. (−1 , 1/2) & (1/2 , 1) Hence, f(x) is strictly decreasing for 𝑥 ∈ (−1 , 1/2) & f(x) is strictly increasing for 𝑥 ∈ (1/2, 1) Hence, f(𝑥) is neither decreasing nor increasing on (−𝟏 , 𝟏). Hence Proved