Question

Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x-values at which they occur

Answer 1

The absolute maximum of the function is 184 and the minimum value of the function is -72.

What is the absolute maximum value?

If the graph of an absolute value function opens downward, the y-value of the vertex is the maximum value of the function.

Given the function f(x) = x⁴-18x²+9 at the interval [-5, 5], the absolute maximum and minimum values at this endpoints are as calculated;

At end point x = -5

f(-5) = (-5)⁴-18(-5)²+9

f(-5) = 625-450+9

f(-5) = 184

At end point x = 5

f(5) = (5)⁴-18(5)²+9

f(5) = 625-450+9

f(5) = 184

To get the critical point, this point occurs at the turning point i.e at

dy/dx = 0

if y = x⁴-18x²+9

dy/dx = 4x³-36x = 0

4x³-36x = 0

4x (x²-9) = 0

4x = 0

x = 0

x²-9 = 0

x² = 9

x = ±3

Using the critical points [0, ±3]

when x = 0, f(0) =  0⁴-18(0)+9

f(0) = 9

Similarly when x = 3, f(±3)=  (±3)⁴-18(±3)²+9

f(±3) = 81-162+9

f(±3) = -72

It can be seen that the absolute minimum occurs at x= ±5 and the absolute minimum occurs at x =±3

absolute maximum = 184

absolute minimum = -72

Hence, the absolute maximum of the function is 184 and the minimum value of the function is -72.

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