Q.No. 50 A function f is given as:
f(X) = 4X − X2
The function f is maximized when X is equal to _____________.
Maximize Quadratic Function f(X) = 4X – X²: Find Exact X Value
The function f(X) = 4X − X² reaches its maximum at X = 2.
Problem Analysis
Rewrite f(X) in standard quadratic form:
f(X) = −X² + 4X.
This is a parabola opening downward since the coefficient of X² is negative, confirming a maximum exists.
Vertex Formula Method
For f(X) = aX² + bX + c with a = −1, b = 4:
X-coordinate of the vertex:
X = −b / (2a) = −4 / (2 × −1) = 2.
Calculus Method
Differentiate: f′(X) = 4 − 2X.
Set to zero: 4 − 2X = 0 ⇒ X = 2.
Second derivative: f′′(X) = −2 < 0 confirms a maximum.
Verification
Substitute X = 2:
f(2) = 4(2) − (2)² = 8 − 4 = 4.
Values nearby confirm peak: f(1) = 3, f(3) = 3.
