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Chapter 3: Problem 43

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)=5 x^{2}-5 x$$

Short Answer

Expert verified
The function \(f(x)=5x^{2}-5x\) has a minimum value. The minimum value is -0.5 and it occurs at \(x = 0.5\). The domain of the function is \(-\infty, \infty\) and the range is \([-0.5, \infty)\).

Step by step solution

01

- Determine the nature of the function

For the given function \(f(x)=5x^{2}-5x\), notice that the coefficient in front of \(x^{2}\) is 5, which is greater than zero. This means that the quadratic graph opens upwards and has a minimum point.
02

- Find the minimum value and where it occurs.

The minimum or maximum occurs at the vertex of the parabola, which is given by \(-b/2a\). For this function, substitute \(a=5\) and \(b=-5\) into the formula to get x-coordinate of the vertex: \(x = -(-5)/(2*5) = 0.5\). Substitute \(x = 0.5\) back into the equation to get the minimum value \(f(0.5) = 5 * (0.5)^{2} - 5 * 0.5 = -0.5\).
03

- Determine the function's domain and range.

The domain of a quadratic function is all real numbers as you can substitute any real number for x into the equation. Therefore, Domain= \(-\infty, \infty\). The range of a quadratic function that opens upwards and possesses a minimum value is \([minimum \, value, \infty)\). Therefore, Range= \([-0.5, \infty)\).

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