Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show that the function is concave upward wherever it is defined. $$ f(x)=4 x^{2}-12 x+7 $$

Step-by-Step Solution

Answer

The function \(f(x) = 4x^2 - 12x + 7\) has a first derivative of \(f'(x) = 8x - 12\) and a second derivative of \(f''(x) = 8\). Since the second derivative is a positive constant, the function is concave upward wherever it is defined.

Step 1: Find the first derivative

To find the first derivative, apply the power rule to each term: $$ f'(x) = \frac{d}{dx}(4x^2 - 12x + 7) $$ $$ f'(x) = 8x - 12 $$

Step 2: Find the second derivative

Now, we have to find the second derivative by taking derivative of the first one, again applying the power rule: $$ f''(x) = \frac{d}{dx}(8x - 12) $$ $$ f''(x) = 8 $$

Step 3: Determine concavity

The second derivative, f''(x), is equal to a constant 8, which is positive. A positive second derivative means that the function is concave upward. Therefore, the function f(x) = 4x^2 - 12x + 7 is concave upward wherever it is defined.

Key Concepts

Second Derivative Test

The second derivative test is a useful tool in calculus for determining the concavity of functions, and by extension, the nature of critical points. When you find the second derivative of a function, it gives you insights into the behavior of the function's graph. In simple terms:- A positive second derivative suggests that a curve is concave upward.- A negative second derivative indicates that the curve is concave downward.For the function \( f(x) = 4x^2 - 12x + 7 \), we follow this test. The second derivative \( f''(x) \) is calculated to be 8. Since 8 is greater than zero, this confirms the function is concave upward throughout its domain. This constant value in the second derivative means the curve doesn't change concavity, maintaining this upward curvature across the whole range.

Power Rule

The power rule is a fundamental technique in calculus used to differentiate functions. It's specifically useful when dealing with polynomial terms. The rule is simple: if \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \). Let's break it down:- It states that you multiply the term by its power.- Then, decrease the power by one.In the exercise, we applied the power rule to find the derivatives of each term in the function \( 4x^2 - 12x + 7 \). For \( 4x^2 \), the derivative was calculated as \( 8x \) because \( 2 \times 4x^{2-1} = 8x \). For the constant \( 7 \), the derivative is zero, as constants disappear when differentiated. Using the power rule simplifies finding the rate of change or slope of a function.

Quadratic Functions

Quadratic functions are those of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). They plot as parabolas on a graph:- If \( a \) is positive, the parabola opens upward.- If \( a \) is negative, it opens downward.The specific function \( f(x) = 4x^2 - 12x + 7 \) is a quadratic function with \( a = 4 \). This positive \( a \) value means the graph of the function is a parabola that opens upwards. When analyzing concavity through derivatives, we find that the constant second derivative confirms its shape. A quadratic function is particularly straightforward as its second derivative is constant. Hence, it's either always concave up or down, depending on the sign of \( a \). Quadratics are commonly used in physics and engineering to estimate basic projectile motions and other real-world parabolic relationships.