Question

Let \( f(x) = a^2x^2 + 2bx + c \) where, a \( \neq \) 0, b, c are real numbers and x is a real variable then

• A. f(x) has no minimum and no maximum
• B. f(x) has a maximum and a minimum
• C. f(x) has a minimum and no maximum
• D. f(x) has a maximum and no minimum

Hint:

The key to determining the extrema of a quadratic function is the sign of the coefficient of the \(x^2\) term. If it's positive, think of a "smiley face" parabola (\(\cup\)), which has a minimum. If it's negative, think of a "frowny face" parabola (\(\cap\)), which has a maximum.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The function \( f(x) = a^2x^2 + 2bx + c \) is a quadratic function. The graph of a quadratic function is a parabola. The existence of a maximum or minimum value depends on the orientation of this parabola, which is determined by the sign of the leading coefficient (the coefficient of the \(x^2\) term).
Step 2: Key Formula or Approach:
For a quadratic function \( F(x) = Ax^2 + Bx + C \):

If the leading coefficient \( A>0 \), the parabola opens upwards. It has a global minimum at its vertex and no maximum.
If the leading coefficient \( A<0 \), the parabola opens downwards. It has a global maximum at its vertex and no minimum.
Step 3: Detailed Explanation:
The given function is \( f(x) = a^2x^2 + 2bx + c \). This is a quadratic function of the form \( Ax^2 + Bx + C \), where:

The leading coefficient is \( A = a^2 \).
The linear coefficient is \( B = 2b \).
The constant term is \( C = c \).
We are given that a, b, and c are real numbers and \( a \neq 0 \). Let's analyze the leading coefficient, \( A = a^2 \). Since a is a non-zero real number, its square, \( a^2 \), must be strictly positive. \[ a^2>0 \] Because the leading coefficient is positive, the graph of the function \( f(x) \) is a parabola that opens upwards. A parabola that opens upwards extends infinitely in the positive y-direction, so it has no maximum value. It has a single lowest point at its vertex, which represents the global minimum value of the function. Therefore, the function f(x) has a minimum and no maximum.
Step 4: Final Answer:
f(x) has a minimum and no maximum.