Ex 6.5,4 - Class 12

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Ex 6.5,4 Prove that the following functions do not have maxima or minima: (i) 𝑓 (𝑥) = ﷐𝑒﷮𝑥﷯ Given 𝑓 (𝑥) = ﷐𝑒﷮𝑥﷯ Finding maxima or minima 𝑓′(𝑥) = ﷐𝑒﷮𝑥﷯ Putting f‘ (x) = 0 𝑒𝑥 = 0 This is not possible for any value of x. ∴ f (x) does not have a maxima or minima. Ex 6.5,4 Prove that the following functions do not have maxima or minima: (ii) g(x) = log x Given g (x) = log x Finding maxima or minima g‘ (x) = ﷐1﷮𝑥﷯ Putting g’ (x) = 0 ﷐1﷮𝑥﷯=0 𝑥 =﷐1﷮0﷯ 𝑥 = 𕔴 This is not defined for x. So, g (x) does not have a maxima or minima. Ex 6.5,4 Prove that the following functions do not have maxima or minima: (iii) ℎ﷐𝑥﷯= ﷐𝑥﷮3﷯+﷐𝑥﷮2﷯+𝑥+1 Given h (x) = x3 + x2 + x + 1 Finding maxima or minima ℎ’ ﷐𝑥﷯ = 3﷐𝑥﷮2﷯+2𝑥+1 Putting ℎ’ ﷐𝑥﷯= 0 3﷐𝑥﷮2﷯+2𝑥+1=0 Here 𝑎 = 3, b = 2, & c = 1 x = ﷐− 2 ± ﷐﷮4 − 4(3)(1)﷯﷮6﷯ x = ﷐− 2 ± ﷐﷮4 − 12﷯﷮6﷯ x = ﷐−2 ± ﷐﷮− 8﷯﷮6﷯ x = ﷐− 2 ± 2﷐﷮− 2﷯﷮6﷯ x = ﷐−𝟏 ± ﷐﷮− 𝟐﷯﷮𝟑﷯ Since root has minus sign, x has no real value ∴ h (x) does not have a maxima of minima