Problem 49 - A Textbook of Differential Calculus

Question

Statement I The equation \(f(x)=4 x^{5}+20 x-9=0\) has only one real root. Statement II \(f^{\prime}(x)=20 x^{4}+20=0\) has no real root.

Step-by-Step Solution

Answer

Both statements are true: \(f(x)\) has one real root, and \(f'(x) = 0\) has no real roots.

Step 1: Analyze Statement II

Statement II asserts that the derivative of the function, given by \(f'(x) = 20x^4 + 20\), has no real root. To check this, set \(f'(x) = 0\):\[20x^4 + 20 = 0.\]Subtract 20 from both sides:\[20x^4 = -20.\]Divide by 20:\[x^4 = -1.\] Since the fourth power of a real number cannot be negative, the equation has no real roots, confirming the statement.

Step 2: Understand Statement I in Context

Statement I claims that the equation \(f(x) = 4x^5 + 20x - 9 = 0\) has only one real root. To investigate this, consider the nature of \(f(x)\). With no real roots for \(f'(x)=0\), the function \(f(x)\) does not change direction and is either always increasing or always decreasing.

Step 3: Determine Monotonicity of \(f(x)\)

Given that \(f'(x) = 20x^4 + 20\) is always positive (since 20 and \(x^4\) are both positive), \(f(x)\) is a monotonically increasing function for all real \(x\). This implies that \(f(x)\) crosses the x-axis only once, supporting the claim that there is only one real root.

Step 4: Conclusion on Statements

Statement II is true as demonstrated by its supporting explanation. Statement I is true, supported by the analysis that \(f(x)\) is monotonic with no turning points due to \(f'(x)\) having no real roots. Together, both statements support that there is only one real root, directly due to the non-existence of local maxima or minima.

Key Concepts

Monotonic Functions

A function is said to be monotonic when it is consistently increasing or decreasing over its entire domain. If a function increases with every increase in the input values, it is called monotonically increasing. Conversely, if it decreases with every increase in the input, it is monotonically decreasing.

Monotonic functions are straightforward because they behave consistently, making them predictable. They do not have local maxima or minima, as they never turn back on themselves.

• An increasing function goes uphill as you move left to right.
• A decreasing function goes downhill as you move left to right.

In the context of our exercise, the function \(f(x) = 4x^5 + 20x - 9\) is demonstrated to be a monotonically increasing function. This is because its derivative, \(f'(x) = 20x^4 + 20\), is always positive. Consequently, this ensures that \(f(x)\) climbs ever higher, never doubling back over the same x-value. Thus, it guarantees that \(f(x)\) passes through the x-axis only once, showing there is one real root.

Roots of Equations

Finding the roots of an equation means determining the values for which the function equals zero. In simpler terms, a function's root is the solution to the equation \(f(x) = 0\). This is where the function intersects the x-axis in a graph.

The process of finding roots involves using various methods - graphical, numerical, or algebraic techniques - to isolate these points where the function equals zero. In polynomial functions, the degree of the polynomial suggests the maximum number of roots.

For the function \(f(x) = 4x^5 + 20x - 9\), determining where it crosses the x-axis involves understanding its behavior. Since the function is monotonic and always increasing, as outlined previously, it ensures either one intersection or none with the x-axis if it begins below or above it. The focus is on those precise values that satisfy \(f(x) = 0\). In this case, there's only one real root since the function doesn’t turn back, confirming the solution of the exercise.

Derivatives

Derivatives are fundamental in calculus, measuring how a function changes as its input changes. The derivative expresses the rate of change or the slope of a function's graph at a particular point. It's a core tool for analyzing the behavior of functions, especially in determining when they are increasing or decreasing.

The derivative of a function \(f(x)\) is often denoted by \(f'(x)\). When you set \(f'(x) = 0\), you are looking for critical points - places where the function might change from increasing to decreasing or vice versa.

In our exercise, the function \(f(x) = 4x^5 + 20x - 9\) has its derivative \(f'(x) = 20x^4 + 20\). Solving \(f'(x)=0\) results in the equation with no real solution (as shown by attempting to set it equal to zero). Therefore, the derivative provides insight into the function's behavior by showing there are no critical points, supporting that \(f(x)\) is monotonic. Understanding this derivative confirms the function's single real root due to its consistent behavior.