Problem 71 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(m\) the function \(m x^{2}-9 m x+5 m+1\) is positive for all \(x ?\)

Step-by-Step Solution

Answer

For the function \(f(x) = mx^2 - 9mx + 5m + 1\) to be positive for all \(x\), the values of m must satisfy \(0 < m < \frac{4}{61}\).

Step 1: 1. Write the quadratic function in standard form

The function is given as \(f(x) = mx^2 - 9mx + 5m + 1\).

Step 2: 2. Calculate the discriminant and apply the condition for no real roots

The discriminant for a quadratic function \(ax^2+bx+c\) is given by \(D = b^2 - 4ac\). For our function, we have \(a = m\), \(b = -9m\), and \(c = 5m + 1\). Now, let's find the discriminant: \(D = (-9m)^2 - 4(m)(5m + 1)\) \(D = 81m^2 - 20m^2 - 4m\) \(D = 61m^2 - 4m\) For the function to have no real roots, the discriminant must be less than zero, i.e., \(D < 0\).

Step 3: 3. Solve the inequality to find the values of m for which the function is positive for all x

We need to solve the inequality \(61m^2 - 4m < 0\). First, let's factor out the common factor of 'm': \(m(61m - 4) < 0\) Now, we can use the properties of inequalities to find the values of m that make the function positive for all x. We have two factors being multiplied together (m and 61m - 4), so one of them must be positive and the other must be negative: - If m > 0, then \(61m - 4 > 0\). Solving this, we get \(m > \frac{4}{61}\). - If m < 0, then \(61m - 4 < 0\). Solving this, we get \(m < \frac{4}{61}\). By considering the leading coefficient, which is 'm', we can determine that the function will be positive when m > 0. Thus, combining these conditions, we get: For the function \(f(x) = mx^2 - 9mx + 5m + 1\) to be positive for all \(x\), the values of m must satisfy \(0 < m < \frac{4}{61}\).

Key Concepts

Discriminant of a Quadratic Equation

The discriminant is a powerful tool in understanding the nature of roots for quadratic equations. A quadratic equation is generally given by the form \(ax^2+bx+c=0\text{, where } a, b, \text{and } c \text{are constants. The discriminant is calculated using the formula } D = b^2 - 4ac\text{.}\)

The value of the discriminant provides critical information:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root (also known as a repeated or double root).
• If \(D < 0\), the equation has no real roots; instead, it has two complex roots.

Understanding the discriminant helps in determining the sign of a quadratic function over an interval, which is crucial when solving inequalities and analyzing function positivity conditions.

Solving Inequalities

Inequalities are solved using principles similar to those used in equations, but with extra attention to the direction of the inequality. When we multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed. This is a key point to remember.

For the given exercise, we approach solving the inequality \(61m^2 - 4m < 0\) by factoring and considering test points or intervals. This process involves determining the intervals where the product of the factors is less than zero. Graphical representations and the use of a sign chart can be very helpful in visualizing these intervals and comprehending the overall behavior of the inequality.

Real Roots of Quadratic Equations

The nature of the roots of quadratic equations is significant in understanding the graph of a quadratic function. Real roots correspond to the x-intercepts of the graph. For \(ax^2 + bx + c = 0\), the value of \(a\) (the leading coefficient) dictates the direction of the parabola opening.

When \(a > 0\), the parabola opens upwards, and when \(a < 0\), it opens downwards. The number and type of real roots greatly influence the function's behavior. In our example, having no real roots indicates that the parabola does not cross the x-axis, and the sign of \(a\) will then determine whether the entire parabola is above or below the x-axis, thereby staying positive or negative for all \(x\).