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CUET
-- Mathematics - Section B1
Applications of Derivatives
The intervals of monotonicity of the function y=x2−loge|x|.(x≠0) |
(−∞,−1√2)∪(0,1√2) (∞,1√2)∪(0,−1√2) (∞,1√2)∪(0,1√2) (−∞,−1√2)∪(0,−1√2) |
(−∞,−1√2)∪(0,1√2) |
Let y=x2−loge|x| f(x)={x2−loge(−x),x<0x2−log(x),x>0 ⇒f′(x)={2x−1(−x)(−1),x<02x−1x,x>0 ∴f'(x)=2x-\frac{1}{x} for all x(x≠0) f'(x)=\frac{2x^2-1}{x} f'(x)=\frac{(\sqrt{2}-1)(\sqrt{2}+1)}{x} Using number line rule, as shown in figure it gives f'(x)>0 when x∈(-\frac{1}{\sqrt{2}},0)(\frac{1}{\sqrt{2}},∞) and f'(x)<0 when x∈(-∞,-\frac{1}{\sqrt{2}})(0,\frac{1}{\sqrt{2}}) ∴ f(x) is increasing when x∈(-\frac{1}{\sqrt{2}},0)(\frac{1}{\sqrt{2}},∞) and decreasing when x∈(-∞,-\frac{1}{\sqrt{2}})(0,\frac{1}{\sqrt{2}}) |