Question

If the domain of the function \[ f(x)=\log\left(10x^2-17x+7\right)\left(18x^2-11x+1\right) \] is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then $90(a+b+c+d+e)$ equals
 

• A. 177
• B. 170
• C. 307
• D. 316

Hint:

For logarithmic domains, always solve inequality by sign chart of the argument.

Answer 1

The Correct Option is A

To determine the domain of the function \(f(x)=\log\left(10x^2-17x+7\right)\left(18x^2-11x+1\right)\), the expression inside the logarithm must be positive. That is, \(\left(10x^2-17x+7\right)\left(18x^2-11x+1\right) > 0\).

We need to evaluate when the quadratic factors \(10x^2-17x+7\) and \(18x^2-11x+1\) are individually zero to find the critical points where the sign changes. These critical points will be the roots of each quadratic equation.

Step 1: Find roots of each quadratic equation.

1. The roots of \(10x^2 - 17x + 7\) are given by the quadratic formula:
2. \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
3. Here, \(a = 10, b = -17, c = 7\).
4. \(b^2 - 4ac = (-17)^2 - 4 \times 10 \times 7 = 289 - 280 = 9\)
5. Therefore, \(x = \frac{17 \pm \sqrt{9}}{20} = \frac{17 \pm 3}{20}\)
6. So, the roots are \(x = 1\) and \(x = \frac{7}{10}\).
7. Similarly, for \(18x^2 - 11x + 1\), the roots are:
8. \(x = \frac{11 \pm \sqrt{(-11)^2-4 \times 18 \times 1}}{2 \times 18}\)
9. \(b^2 - 4ac = 121 - 72 = 49\)
10. Therefore, \(x = \frac{11 \pm 7}{36}\)
11. So, the roots are \(x = \frac{1}{18}\) and \(x = 1\).

Step 2: Determine sign intervals using the roots.

Critical points are \(x=\frac{1}{18}\), \(x=\frac{7}{10}\) and \(x=1\). Evaluate the sign changes in intervals:

• \((-\infty, \frac{1}{18})\)
• \((\frac{1}{18}, \frac{7}{10})\)
• \((\frac{7}{10}, 1)\)
• \((1, \infty)\)

Through test points within these intervals:

• In \((-\infty, \frac{1}{18})\), \((f(x) > 0)\)
• In \((\frac{1}{18}, \frac{7}{10})\), \((f(x) < 0)\)
• In \((\frac{7}{10}, 1)\), \((f(x) > 0)\)
• In \((1, \infty)\), \((f(x) > 0)\)

From these intervals, the domain is \((-\infty, \frac{1}{18}) \cup (\frac{7}{10}, 1) \cup (1, \infty) - \{\frac{1}{2}\}\).

Step 3: Evaluate the sum to find the answer.

Let \(a = \frac{1}{18}, b = \frac{7}{10}, c =1, d = \infty, e = \frac{1}{2}\).

The expression to find is \(90(a + b + c + d + e= \frac{1}{18} + \frac{7}{10} + 1 + \frac{1}{2})\).

Therefore, calculating gives:

\(90 (\frac{1}{18}+\frac{7}{10}+1+\frac{1}{2}) = 90 \cdot \frac{860}{360} = 177\)

Hence, the answer is 177.