Given f(x) = 2x(x-2)^3 (a) Find the first and second derivative and factor compl
ID: 3341533 • Letter: G
Question
Given f(x) = 2x(x-2)^3(a) Find the first and second derivative and factor completely.
(b) Find critical points of the first and second derivative
(c) Create a separate sign chart for each derivative (indicating intervals where each derivative is positive and negative)
(d) State intervals where the function is increasing and decreasing.
(e) State the relative maxand min values for the function (give exact values for x and y).
(f) State the intervals where the function is concave up and concave down.
(g) State the inflection points for the function. (give exact values for x and y) Given f(x) = 2x(x-2)^3
(a) Find the first and second derivative and factor completely.
(b) Find critical points of the first and second derivative
(c) Create a separate sign chart for each derivative (indicating intervals where each derivative is positive and negative)
(d) State intervals where the function is increasing and decreasing.
(e) State the relative maxand min values for the function (give exact values for x and y).
(f) State the intervals where the function is concave up and concave down.
(g) State the inflection points for the function. (give exact values for x and y) Given f(x) = 2x(x-2)^3
(a) Find the first and second derivative and factor completely.
(b) Find critical points of the first and second derivative
(c) Create a separate sign chart for each derivative (indicating intervals where each derivative is positive and negative)
(d) State intervals where the function is increasing and decreasing.
(e) State the relative maxand min values for the function (give exact values for x and y).
(f) State the intervals where the function is concave up and concave down.
(g) State the inflection points for the function. (give exact values for x and y)
Explanation / Answer
f(x) = 2x(x-2)^3
a)
f'(x) = 2*(x-2)^3 + 2x * 3*(x-2)^2
= (x-2)^2 * [2x - 4+ 6x]
= (x-2)^2 * [8x - 4]
f''(x) = 2 *(x-2) * [8x - 4] + (x-2)^2 * [8]
= (x-2) * [16x - 8 + 8x - 16]
= (x-2) *(24x - 24)
b)
f'(x) = 0
==>
(x-2)^2 * [8x - 4] = 0
==>
x = 2, 1/2
critical points are (2, 0) and (1/2, -27/8)
and
f''(x) = 0
==>
(x-2) *(24x - 24) = 0
==>
x = 2, 1
critical points are (2, 0) and (1, -2)
D)
f'(2) = 0
f'(1) = 4
f'(0) = 4 * -4 = -16
f'(1/2) = 0
f'(-1) = -108
so, it increases on (0, +infy)
and
decreases on (-infy, 0)
E) it has no relative minima and maxima.
F)
f''(x) = (x-2) *(24x - 24)
f''(2) = 0
f''(3/2) = -6 <0
f''(1) = 0
f''(1/2) = 18
f''(0) = 48 > 0
f''(3) = 48 >0
so, it is concave up on (-infy, 1) U (2, +infy)
and concave down on (1, 2)
G)
inflection points are (1, 0) and (2, 0)
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