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Chapter 3: Problem 44

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. $$f(x)=6 x^{2}-6 x$$

Short Answer

Expert verified
a. The function has a minimum value. b. The minimum value is -1.5 and it occurs at \(x = 0.5\). c. The domain of the function is \(D: -\infty < x < \infty\) and its range is \(R: -1.5 \leq y < \infty\).

Step by step solution

01

Determine the type of value

The coefficient of the \(x^2\) term is 6 which is positive. Therefore, the function has a minimum value.
02

Find the minimum value and its location

The minimum or maximum value occurs at the vertex of the parabola. The x-coordinate of the vertex is given by \(-b/2a\). Substituting \(a = 6\) and \(b = -6\) into this formula gives \(-(-6)/2*6 = 0.5\). Substitute \(x = 0.5\) into the function to find the minimum value. \(f(0.5) = 6*(0.5)^2 - 6*0.5 = -1.5\). Hence the function's minimum value is -1.5 occurring at \(x = 0.5\).
03

Identify the domain and range of the function

The domain of any quadratic function is all real numbers, hence \(D: -\infty < x < \infty\). Since the function has a minimum value, the range is from that minimum value to positive infinity. Hence, \(R: -1.5 \leq y < \infty\).

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