Answer:
absolute maximum = 184
absolute minimum = -72
Explanation:
Given the function f(x) = x⁴-18x²+9 at the interval [-5, 5], the absolute maximum and minimum values at this end points 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 points 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 and absolute minimum occurs at x =±3
absolute maximum = 184
absolute minimum = -72