Absolute Extrema of a Function

4.1: Absolute Extrema of a function . f- 1×7=1/2 F) [° ' ✗ E(o,2]- Examples: a- 1. Find the absolute maximum and minimum values of 𝑓(𝑥 ) = 𝑥 3 − 3𝑥 + 2 over a) [-2, 3/2] b) [-3, -3/2]. ' f- ( X ) ' f- ( X) 3×2-3 = = - 0 31×2-13=0 critical values offend over y 1 , -1 , -2 . § . • : 1¥ the points absolute valve of ✗ . sis : × . men y 1 , -1 [-31-1] ☒É¥ - × : = 1 , -1 ✗= : [-2/3] over ✗ -3 . : abs min 1312,5¥ -1¥ . value off . abs max . . off y ✗ = 8 § - } 16.x C- 1,87 . 2 2. Find the absolute extrema of 𝑓(𝑥) = 𝑥 3 − 16𝑥 3 over the interval [-1, 8]. f= § ✗ § = % - ✗ 16 % 32g - § ¥ [ = . § = ' ¥ ( , %↳ 2- values at ✗ flex, fails to exist ¥ , ✗ f- 1×1=0 - - ✗ ,, . I} 2g . u - ] 4) ✗ 2-4=0 where ✗ at ✗ - = 2 ✗ fix, values : f M.us 0 : of 2 o F. the point of absolute value mi : ✗= z -2 value where ✗ critical , 0 2, , does not - ¥-0 =0 -2 belong to fi , 8) If 8 I - - ✗ is ÷ the point of absolute value max . 3. 𝑓(𝑥) = 4𝑥 3 + 3𝑥 2 − 6𝑥 − 5 over [-2, 1]. f- 12×2-16×-6=0 1=0 2×2+11-1=0 ✗ ± ✗ f-1×7 = ÷ . -675 | = ' -13 _i±E = -1¥ -4 absolute minimum absolute Max 4. 𝑓(𝑥 ) = −𝑥 3 + 𝑥 2 + 5𝑥 − 1 over (0, ∞). 5. 𝑓(𝑥 ) = 5𝑥 + f- f ' 35 𝑥 over (0, ∞). 4×7=5-3×5-2 / 5×2-35=0 ✗7=0 - IT 4- Coin ) ✗ = JT -57 , There since f- is " only ( X) and f "W = ) ✗ value where ¥3 > = critical one IT 6. 𝑓(𝑥 ) = (𝑥 + 1)1⁄3 𝑜𝑣𝑒𝑟 [−2, 26]. 0 ¢-7 y , = , ✗ =J> 26.45 ) 26.45 is a point of relative absolute) min and ( value of fcx) .