what is the absolute maximum

Question

If f(x)= ((1/4)x⁴)+((1/3)x³)-(2x²)-(4x)+6 for x in the interval [–4,10], then f has an absolute maximum at ______.

Answer 1

To find the maximum, we use the first derivative test.  Set the derivative of f(x) equal to zero.

 

f'(x) = 0

 

x³ + x² - 4x - 4 = 0

 

 

Use synthetic division to factor the left side of the equation.

 

(x + 2)(x² - x - 2) = 0

 

(x + 2)(x - 2)(x + 1) = 0

 

 

x = -2

x = -1

x = 2

 

 

These x values are the critical points, also known as the location of the maximum/minimum value. 

 

Next, we need to use test points to see which of these x values is a maximum.  We can use  x=-3 , x=-1.5, x=1, and x=3, , can plug them into the derivative.  If the derivative goes from positive to negative between the x value, then we have a maximum.

 

 

f'(-3) = (-3 + 2)(-3 - 2)(-3 + 1)

        = (-1)(-5)(-2)

        = -10

 

f'(-1.5) = (-1.5 + 2)(-1.5 - 2)(-1.5 + 1)

           = (0.5)(-3.5)(-0.5)

           = 0.875

 

f'(1) = (1 + 2)(1 - 2)(1 + 1)

       = (3)(-1)(2)

       = -6

 

f'(3) = (3 + 2)(3 - 2)(3 + 1)

       = (5)(1)(4)

       = 20

 

The local maximum lies at x = -1.   Now to find the global maximum, we need to evaluate 2 other points:

 

f(-4)   and   f(10) 

 

This is because we need to find the points at the end intervals and compare it to the local maximum, which is f(-1).  The value that has the highest f(x) value is the global maximum.

 

I leave this part to you.