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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.
Learn more about absolute maximum value here;
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