Let f(x)=4 x^2-4 a x+a^2-2 a+2 and the global minimum value of f(x) for x ∈[0,2] is equal to 3 .

step 1 So, let's solve this question. So in this question, we have F of X is equal to 4x square minus 4

Let f(x)=4 x^2-4 a x+a^2-2 a+2 and the global minimum value...

Let $f(x)=4 x^{2}-4 a x+a^{2}-2 a+2$ and the global minimum value of $f(x)$ for $x \in[0,2]$ is equal to 3 .
The values of a for which $f(x)$ is monotonic for $x \in[0,2]$ are
(A) $a \leq 0$ or a $\geq 4$
(B) $0 \leq a \leq 4$
(C) $a>0$
(D) None of these

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form f(x) = ax² + bx + c. Their graph is a parabola which opens upward if a > 0 and downward if a < 0. Understanding the basic structure of quadratics, including the vertex form and axis of symmetry, is essential for analyzing properties like the minimum or maximum values over a specified domain.

Monotonicity

Monotonicity refers to a function either being entirely nonincreasing or nondecreasing over an interval. For a function to be monotonic on a given interval, it must consistently increase, decrease, or remain constant without any local turns. This property is often checked using the first derivative of the function, ensuring that the derivative does not change sign over the interval.

Derivative Test for Monotonicity

The derivative test is used to determine if a function is monotonic over an interval. By calculating the derivative of the function and examining its sign at different points within the interval (specifically at the endpoints), one can decide whether the function is consistently increasing or decreasing. If the derivative is nonnegative throughout, the function is nondecreasing; if it is nonpositive, the function is nonincreasing.

Parameter Analysis in Functions

When a function depends on a parameter, such as a in quadratic functions, the behavior of the function (like its monotonicity or the location of its minimum or maximum) can change with different parameter values. Analyzing the derivative with respect to both the variable and the parameter is crucial to determine the conditions on the parameter that yield certain properties of the function, like ensuring monotonic behavior over a specified interval.

Please give Ace some feedback