Choose the one alternative that best completes the statement or answer the quest

Question

Choose the one alternative that best completes the statement or answer the question. Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval, and indicated the x -value at which they occur. f(x) = x^3_1/2 - 2x + 4; [0, 2] Absolute maximum = 3.2 at x = 0; absolute minimum = 2.5 at x = 2 Absolute maximum = 6 at x = 0; absolute minimum = 3.2 at x = 1 Absolute maximum = 4 at x = 2; absolute minimum = 2.5 at x = 1 Absolute maximum = 6 at x = 2; absolute minimum = 2.5 at x = 1

Explanation / Answer

f(x) = x^3-0.5*x^2 -2x+4
f'(x) = 3x^2-x-2
put f'(x) = 0
3x^2-x-2 = 0
3x^2-3x+2x-2 = 0
3x(x-1)+2(x-1)=0
(3x+2)(x-1)=0
x=1 is only root in [0,2]

so critical poiints are 0,1,2

f''(x) = 6x-1
at x=1 , f''(x) = 5 which is positive
so x=1 is minima

f(1) = 1^3-0.5*1^2 -2*1+4
= 1 - 0.5 -2+4
= 2.5

f(0) = 0^3-0.5*0^2 -2*0+4
= 0 - 0 -0+4
= 4

f(2) = 2^3-0.5*2^2 -2*2+4
= 8 - 2 -4+4
= 6

minima at x=1, and value is 2.5
maxima at x=2 and value is 6
Answer: D

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---