some differential calculus qs?

Question

1.
f(x)= 4x^3-6x^2-144x+1

decreasing on what interval?
increasing on the (MINF, ?) and (?,INF) interval

local max?
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2.
f(x)=4x+2x^-1
local max?
value?

local min?
value?

Answer 1

1. look for min and max values so take the derivative and set equal to 0
df/dx= 12x^2 - 12x - 144 = 0
x^2 - x - 12 = 0
(x - 4)(x + 3) = 0 so x = 4 and x = -3

Take the second derivative for min and max:
d^2f/dx^2 = 24x - 12

at x = 4 ..... d^2f/dx^2 > 0 so a minimum
at x = -3 .... d^2f/dx^2 < 0 so a maximum

Decreasing on (-3,4)
Increasing on (MINF, -3) and (4, INF)

Local max is where x = -3 and y = 271

2. Function is asymptotic at x =0

df/dx = 4 - 2/x^2 = 0 for max and min
4x^2 = 2 and x = +/- SQRT(1/2)

d^2f/dx^2 = 4/x^3

At x = -SQRT(1/2) ...... d^2f/dx^2 < 0 so a maximum
At x = SQRT(1/2) ....... d^2f/dx^2 > 0 so a minimum
Maximum: x = -SQRT(1/2) and y = -4SQRT(2)
Minimum: x = SQRT(1/2) and y = 4SQRT(2)