A. Find an upper bound and a lower bound for each of the following polynomial functions

Question

<!--[if !supportLists]-->1.     <!--[endif]-->y  = 2x³ – x² + 38x – 10

 

<!--[if !supportLists]-->2.     <!--[endif]-->y = 2x⁴ – 3x³ + 6x² + 2x + 16

 

<!--[if !supportLists]-->3.     <!--[endif]-->y = 3x³ – 5x² + x – 4

 

<!--[if !supportLists]-->4.     <!--[endif]-->y = x⁵ + 8x³ + 2x² – 10

 

<!--[if !supportLists]-->5.     <!--[endif]-->y = x⁵ + x⁴ + 5x³ – x - 10

Answer 1

Q 1,3,4,5) Since it is a polynomial with odd degree with positive leading coefficient so the upper bound is positive infinity and lower bound is negative infinity.

2) 2x⁴ – 3x³ + 6x² + 2x + 16

first derivative of the polynomial is 8x³ - 9x² +12x +2

The minimum is when first derivative is 0.

So, 8x³ - 9x² +12x +2 = 0

x = -0.148 (approx) using Newton-Raphson method, if you wish to know how I came up with it, post a new question.

2nd derivative test to confirm that is really a minima.

derivative of 8x³ - 9x² +12x +2 is 24x² -18x +12

when x = -0.148 then 24x² -18x +12 > 0

So the lower bound is when x = -0.148 approx

and upper bound is positive infinity, since it is an even degree polynomial with positive coefficien.