Example 40 - Find absolute max, min values of f(x) = 12x4/3 - Examples

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Example 40 Find absolute maximum and minimum values of a function f given by f (𝑥) = ﷐12 𝑥﷮﷐4﷮3﷯﷯ – ﷐ 6𝑥﷮﷐1﷮3﷯﷯ , 𝑥 ∈ [ – 1, 1] f (𝑥) = ﷐12 𝑥﷮﷐4﷮3﷯﷯ – ﷐ 6𝑥﷮﷐1﷮3﷯﷯ Step 1: Finding f’﷐𝑥﷯ f’﷐𝑥﷯=﷐𝑑﷐12﷐𝑥﷮﷐4﷮3﷯﷯ − 6﷐𝑥﷮﷐1﷮3﷯﷯﷯﷮𝑑𝑥﷯ = 12 × ﷐4﷮3﷯﷐𝑥﷮﷐4﷮3﷯ −1﷯−6 × ﷐1﷮3﷯﷐𝑥﷮﷐1﷮3﷯ −1﷯ = 4 × 4 ﷐𝑥﷮﷐4 − 3﷮3﷯﷯ −2﷐𝑥﷮﷐1 − 3﷮3﷯﷯ = 16 ﷐𝑥﷮﷐1﷮3﷯﷯ −2﷐𝑥﷮﷐−2﷮3﷯﷯ = 16 ﷐𝑥﷮﷐1﷮3﷯﷯ − ﷐2﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ = ﷐16﷐𝑥﷮﷐1﷮3﷯﷯ × ﷐𝑥﷮﷐2﷮3﷯﷯ − 2﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ = ﷐16﷐𝑥﷮﷐1﷮3﷯ + ﷐2﷮3﷯﷯ −2﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ = ﷐16﷐𝑥﷮﷐3﷮3﷯﷯ −2﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ = ﷐16𝑥 − 2﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ = ﷐2﷐8𝑥 − 1﷯ ﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ Hence, f’﷐𝑥﷯=﷐2﷐8𝑥 − 1﷯﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ Step 2: Putting f’﷐𝑥﷯=0 ﷐2﷐8𝑥 − 1﷯﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯=0 2﷐8𝑥−1﷯=0 ×﷐𝑥﷮﷐2﷮3﷯﷯ 2﷐8𝑥−1﷯=0 8𝑥−1= 0 8𝑥=1 𝑥=﷐1﷮8﷯ Note that: Since f’﷐𝑥﷯=﷐2﷐8𝑥 − 1﷯﷮﷐𝑥﷮﷐2﷮3﷯﷯﷯ f’﷐𝑥﷯ is not defined at 𝑥= 0 𝑥=﷐1﷮8﷯ & 0 are critical points Step 3: We are given interval ﷐−1 , 1﷯ Hence calculating f﷐𝑥﷯ at 𝑥=0,﷐1﷮8﷯, −1 , 1 Hence, Absolute maximum value of f(x) is 18 at 𝒙 = –1 & Absolute minimum value of f(x) is ﷐−9﷮4﷯ at 𝒙 = ﷐𝟏﷮𝟖﷯

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