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Algebraic Expressions Exercises

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1. Find the absolute extrema of the function 𝑓ሺ𝑥ሻ = 3𝑥 4 − 4𝑥 3 on the interval ሾ−1,2ሿ To find extreme points, we find and set 𝑓 ′ ሺ𝑥ሻ to 0: 𝑓 ′ ሺ𝑥ሻ = 12𝑥 3 − 12𝑥 2 = 12𝑥 2 ሺ𝑥 − 1ሻ 2 𝑓 ′ ሺ𝑥ሻ = 0 → 12𝑥 2 ሺ𝑥 − 1ሻ = 0 → {12𝑥 = 0 → 𝑥 = 0 𝑥−1=0→𝑥 =1 Both points lie on the interval ሾ−1,2ሿ 𝑓ሺ−1ሻ = 3ሺ−1ሻ4 − 4ሺ−1ሻ3 = 7 Finding 𝑓ሺ𝑥ሻ for endpoints: { 𝑓ሺ2ሻ = 3ሺ2ሻ4 − 4ሺ2ሻ3 = 16 Finding 𝑓ሺ𝑥ሻ for extreme points: { ∴{ 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑀𝑎𝑥: ሺ2,16ሻ 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑚𝑖𝑛: ሺ1, −1ሻ 𝑓ሺ0ሻ = 3ሺ0ሻ4 − 4ሺ0ሻ3 = 0 𝑓ሺ1ሻ = 3ሺ1ሻ4 − 4ሺ1ሻ3 = −1 3 2. Verify directly that 𝐹ሺ𝑥ሻ = 𝑥 2 𝑒 𝑥 , 𝐺ሺ𝑢ሻ = 𝑢2 ln 𝑢 + 𝑢2 , 𝑎𝑛𝑑 𝐻ሺ𝑤ሻ = ξ𝑤 4 + 2𝑤 are the 3 antiderivatives of 𝑓ሺ𝑥ሻ = 𝑥ሺ2 + 3𝑥 3 ሻ𝑒 𝑥 , 𝑔ሺ𝑢ሻ = 𝑢ሺ3 + 2 ln 𝑢ሻ, 𝑎𝑛𝑑 ℎሺ𝑤ሻ = 2𝑤 3 +1 ξ𝑤 4 +2𝑤 , respectively. To test, we’ll find the derivative of 𝐹ሺ𝑥ሻ, 𝐺ሺ𝑢ሻ, 𝑎𝑛𝑑 𝐻ሺ𝑤ሻ. If their derivatives match 𝑓ሺ𝑥ሻ, 𝑔ሺ𝑢ሻ, 𝑎𝑛𝑑 ℎሺ𝑤ሻ respectively, then we have verified directly that 𝐹ሺ𝑥ሻ, 𝐺ሺ𝑢ሻ, 𝑎𝑛𝑑 𝐻ሺ𝑤ሻ are the antiderivatives of 𝑓ሺ𝑥ሻ, 𝑔ሺ𝑢 ...
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statistics word problem , statistics homework help

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statistics word problem , statistics homework help

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