Question

f(x)=x^3-4x^2+5x-2

Find all critical numbers of the function, then use the second derivative test on each critical number to determine if it is a local maximum or minimum. Show your work.

Answer 1

f(x) = x³ - 4x² + 5x - 2

f'(x) = 3x² - 8x + 5  

put f'(x) = 0  

3x² - 8x + 5 = 0  

3x² - 3x - 5x + 5 = 0

3x(x -1) - 5(x - 1) = 0  

(3x - 5)(x -1) = 0

3x - 5 = 0 or x - 1 = 0

x = 5/3 or x = 1  

critical numbers are  

x = 5/3

x = 1

Second derivative test

f'(x) = 3x² - 8x + 5  

f''(x) = 6x - 8

at x = 1

f''(1) = 6(1) - 8 = 6 - 8 = - 2 < 0

at x = 5/3

f''(5/3)   = 6(5/3) - 8 = 10 - 8 = 2 > 0

Thus x = 1 is point of local maximum or at x =1 f(x) is local maximum and x = 5/3 is point of local minimum or f(x) is local minimum at x = 5/3