Math Problem - Need help

Question

I'm working on a math problem and could really use some assistance. Here it is:

Problem: Given the function f(x)=x3−6x2+9x−4f(x)=x3−6x2+9x−4, find the local maxima and minima using the first and second derivative tests.

I would appreciate a step-by-step explanation of how to solve this!

Answer 1

f(x) = x^3 -6x^2 +9x -4

f'(x) = 3x^2 -12x +9 = 0 at turning points, the local extrema, relative max and min

divide by 3

x^2 -4y +3 = 0

factor

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

x = 1 and 3

at x =1, y = 1-6+9-4 = 0, the point (1,0) the relative minimum

at x=3 y = 27-54 +27 -4 = 4, the point (3,4) the relative maximum

f"(x) = 2x -4

f"(1) = 2-4=-2 <0 which means a minimum as the curve is concave up, convex down

f"(3) = 6-4 = 2 >0 which means a maximum as the curve is concave down or convex up

Answer 2

Visit this Desmos link for the graph referenced in the video. Use the slider on point P to see different tangent lines. The graph also displays the slope of the tangent line as you move point P to different locations.

desmos.com/calculator/syqyry3c6p