Finding the Value of b for Minimum Function Value

In this article, we will explore the problem of finding the value of b such that the minimum value of the function f(x) x^2 - 5x b is zero. We will cover the mathematical reasoning and solution step-by-step, while also providing an intuitive understanding of the concepts involved.

Understanding the Function

First, let's understand the function we are working with: f(x) x^2 - 5x b. This is a quadratic function, and one of its key characteristics is that it forms a parabola. Since the coefficient of x^2 is positive (1), the parabola opens upwards, meaning it has a minimum value.

Determining the Minimum Point

The minimum value of a quadratic function occurs at the vertex of the parabola. The x-coordinate of this vertex can be found using the vertex formula:

x -frac{b}{2a}

Here, a and b represent the coefficients from the quadratic function ax^2 bx c. For our function, a 1 and b -5. Substituting these values into the vertex formula, we get:

x -frac{-5}{2 cdot 1} frac{5}{2}

Calculating the Minimum Value

Now that we know the x-coordinate of the vertex, we can substitute this value into the function to find the minimum value:

fleft(frac{5}{2}right) left(frac{5}{2}right)^2 - 5 left(frac{5}{2}right) b

Let's break it down step-by-step:

left(frac{5}{2}right)^2 frac{25}{4} -5 left(frac{5}{2}right) -frac{25}{2} -frac{50}{4}

Substituting these values into the function:

fleft(frac{5}{2}right) frac{25}{4} - frac{50}{4} b -frac{25}{4} b

We want this minimum value to be zero, so:

-frac{25}{4} b 0

Solving for b:

b frac{25}{4}

Therefore, the value of b that satisfies the condition is boxed{frac{25}{4}}.

Verification and Additional Insights

To verify our result, we can take the derivative of the function and set it to zero. The first derivative of f(x) x^2 - 5x b is:

f'(x) 2x - 5

Solving for x when f'(x) 0 gives:

2x - 5 0 implies x frac{5}{2}

This is consistent with our earlier calculation. Additionally, the second derivative:

f''(x) 2

Since f''(x) > 0, this confirms that the function has a minimum at x frac{5}{2}.

Conclusion

In conclusion, the value of b that ensures the minimum value of the function f(x) x^2 - 5x b is zero is frac{25}{4}. This has been verified through the method of finding the vertex of the parabola and the use of derivatives to locate the extremum of the function. Understanding these concepts is crucial for advanced calculus and applications in various fields of mathematics.

Keywords: minimum value function, quadratic function, vertex formula, calculus, optimization