F(x)=3x^4-x^3+4x-2

Question

For the polynomial shown below, find f(-1).
F(x)=3x^4-x^3+4x-2

 

Identify the relative minimum value for the function shown below.
G(x)=x^3-3x^2+2

Answer 1

F(-1) = 3(-1)⁴ - (-1)³ + 4(-1) - 2 = 3 + 1 - 4 - 2 = -2

 

 

 

To find the minimum set the first derivative equal to zero so solve for the critical x values (relative minimum, relative maximum, or inflection points).  Then check that the second derivative is greater than 0 for the relative minimum.

 

First derivative:  G'(x) = 3x² - 6x

 

Second derivative:  G"(x) = 6x

 

 

Minimums occur when G'(x) = 0 and G"(x) > 0.

 

3x(x - 2) = 0

 

x = 0 or x = 2 for possible relative minimums.

 

 

G'(x) = 0 and G"(x) > 0 when x = 2.  The relative minimum would be at x = 2.