Problem 21 - Calculus for JEE ADVANCED

Question

We have \(f(x)=\left[\log _{10}\left(\frac{5 x-x^{2}}{4}\right)\right]^{1 / 2}\) From (1), clearly, \(f(x)\) is defined for those values of \(x\) for which $$ \log _{10}\left[\frac{5 x-x^{2}}{4}\right] \geq 0 $$

Step-by-Step Solution

Answer

\( f(x) \) is defined for \( x \in [1, 3] \).

Step 1: Understand the Problem Requirements

We need to find the values of \( x \) for which \( \log_{10}\left(\frac{5x-x^{2}}{4}\right) \geq 0 \). This requires the term inside the logarithm to be positive and at least 1. A logarithm of any number less than 1 would yield a negative result, contradicting the requirement we have.

Step 2: Express the Inequality Inside the Logarithm

For \( \log_{10}\left(\frac{5x-x^{2}}{4}\right) \geq 0 \), the expression \( \frac{5x-x^{2}}{4} \) must be greater or equal to 1 since \( \log_{10}(1) = 0 \). So we set the inequality: \( \frac{5x-x^{2}}{4} \geq 1 \).

Step 3: Clear the Fraction

Multiply both sides of the inequality \( \frac{5x - x^2}{4} \geq 1 \) by 4 to eliminate the fraction: \( 5x - x^2 \geq 4 \).

Step 4: Rearrange into Standard Quadratic Form

Rearrange the inequality into the standard quadratic form: \( -x^2 + 5x - 4 \geq 0 \), or equivalently \( x^2 - 5x + 4 \leq 0 \).

Step 5: Solve the Quadratic Equation

Find the roots of the equation \( x^2 - 5x + 4 = 0 \) using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). Here, \( a = 1, \ b = -5, \ c = 4 \). Calculate the discriminant \( (5)^2 - 4 \cdot 1 \cdot 4 = 1 \). The roots are \( x = \frac{5 \pm 1}{2} \), which gives us \( x = 3 \) and \( x = 1 \).

Step 6: Determine the Intervals

Now we examine the intervals determined by the roots: \( (-\infty, 1) , (1, 3) , (3, \infty) \). We test a point from each interval to determine where the inequality \( x^2 - 5x + 4 \leq 0 \) holds true: Test \( x = 0, x = 2, x = 4 \).

Step 7: Test Intervals

For the interval \((-\infty, 1)\), pick \(x = 0\): \(0^2 - 5(0) + 4 = 4 > 0\) (does not satisfy). For \((1, 3)\), pick \(x = 2\): \(2^2 - 5(2) + 4 = 0\) (satisfies). For \((3, \infty)\), pick \(x = 4\): \(4^2 - 5(4) + 4 = 0 > 0\) (does not satisfy).

Step 8: Conclusion

The inequality \( x^2 - 5x + 4 \leq 0 \) is satisfied in the interval \([1, 3] \). Therefore, \( f(x) \) is defined for \( x \in [1, 3] \).

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential part of algebra and offer a way to express exponential relationships. Simply put, a logarithm answers the question: "To what power must a base number be raised, to produce another number?" In the expression \( \log_{10}(x) \), the base is 10, which is common in scientific and financial calculations.

Logarithms follow a few key rules:

• The logarithm of 1 to any base is always 0, since any number raised to the power of 0 equals 1. Thus, \( \log_{10}(1) = 0 \).
• The logarithm of a base raised to its own power returns that power itself, so \( \log_{10}(10) = 1 \).
• They are undefined for negative numbers and zero, since no real number x exists such that 10 raised to it gives a negative number or zero.

Understanding these rules helps when examining the domain of functions involving logarithms. In the function given, \( f(x) = \left[ \log_{10}\left(\frac{5x-x^{2}}{4}\right) \right]^{1/2} \), the expression inside the logarithm needs to be positive and at least 1 to keep the logarithm non-negative, since \( \log_{10}(\text{something less than 1}) \) becomes negative. This ensures the function is well-defined over its domain.

Quadratic Equations

Quadratic equations are fundamental in mathematics and appear in various forms of analysis and graphing. A standard quadratic equation is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). In our exercise, we encountered rearranging into a quadratic form: \( x^2 - 5x + 4 \leq 0 \).

These equations are typically solved using:

• Factoring: If the quadratic can be factored into \( (px + q)(rx + s) = 0 \), we can find the roots \( x \) by setting each factor to zero.
• Quadratic Formula: When factoring is complex, apply the formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
• Completing the Square: This transforms the quadratic into a perfect square trinomial, making it easier to handle.

For our specific quadratic \( x^2 - 5x + 4 = 0 \), we used the quadratic formula to find the roots: \( x = 3 \) and \( x = 1 \). These roots help us delineate intervals where the inequality \( x^2 - 5x + 4 \leq 0 \) holds true, which is crucial for determining the function's domain.

Domain of a Function

The domain of a function refers to all possible input values (usually \( x \)) which make the function work without any mathematical errors, such as dividing by zero or taking a logarithm of a negative number. In the function \( f(x) = \left[ \log_{10}\left(\frac{5x-x^{2}}{4}\right) \right]^{1/2} \), specifying the domain is critical as it involves a logarithm.

To define the domain here, we must ensure:

• The expression inside the logarithm \( \frac{5x-x^{2}}{4} \) is greater than or equal to 1, ensuring the logarithm is non-negative.
• The expression is always positive; otherwise, the logarithm itself becomes undefined.

Solving these conditions led us to form the quadratic inequality \( x^2 - 5x + 4 \leq 0 \), which indicates where the given expression remains valid. This inequality, after testing various intervals generated by its roots, showed that the function is well-defined for the interval \([1, 3]\). This knowledge allows us to understand for which \( x \) values the function can accurately be evaluated, ensuring it operates smoothly without mathematical inconsistencies.