Problem 15 - The Calculus with Analytic Geometry

Question

(a) Draw a sketch of the graph of the given function on the indicated interval; (b) test the three conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem and determine which conditions are satisfied and which, if any, are not satisfied; and (c) if the three conditions in part (b) are satisfied, determine a point at which there is a horizontal tangent line. $$ f(x)=x^{4 / 3}-3 x^{1 / 3},[0,3] $$

Step-by-Step Solution

Answer

Using Rolle's theorem find that \( x = 3/4 \).

Step 1: Sketch the Graph

To draw the graph of the function, analyze the function: \( f(x) = x^{4/3} - 3x^{1/3} \). Evaluate the function at a few notable points in the interval \([0, 3]\).- At \( x = 0 \), `f(0) = 0^{4/3} - 3(0)^{1/3} = 0 - At \( x=1 \), `f(1) = 1^{4/3} - 3(1^{1/3}) = 1 - 3 = -2 - At \( x = 3 \), `f(3) = 3^{4/3} - 3(3^{1/3})`. Evaluate this using the cube roots and exponentiation.

Step 2: Test Condition (i)

Check if \( f \) is continuous on the closed interval \([0,3]\): Since \( x^{4/3} \) and \( -3x^{1/3} \) are continuous, their sum, \( f(x) = x^{4/3} - 3x^{1/3} \), is also continuous.

Step 3: Test Condition (ii)

Check if \( f \) is differentiable on the open interval \((0,3)\): For \( f(x) = x^{4/3} - 3x^{1/3} \), its derivative is \( f'(x) = (4/3)x^{1/3} - (1/x) \). Check if \( f'(x) \) is continuous on \( (0, 3) \). Note: \( f'(0) \) is not defined, so the function is not differentiable at \( x = 0 \). However, \( f(x) \) is differentiable in \( (0, 3) \).

Step 4: Test Condition (iii)

Verify if \( f(0) = f(3) \): Calculate \( f(3) = 3^{4/3} - 3(3^{1/3})\). Since it equals \( f(0) \). Therefore, the third condition is met.

Step 5: Find the Horizontal Tangent Line

Obtain the derivative \( f'(x) = \frac{4}{3}x^{1/3} - x^{-2/3} \). Set the derivative equal to 0 and solve for \( x \): \[ \frac{4}{3}x^{1/3} - x^{-2/3} = 0 \]. Rearrange and solve: \[ \frac{4}{3}x = 1 \]. Find \( x \) value for which derivative equals zero

Key Concepts

Graph Sketching

To understand the behavior of a function, sketching its graph provides a visual insight. The function given in the exercise is: \[ f(x) = x^{4/3} - 3x^{1/3} \] When sketching the graph:

• Evaluate the function at various points in the interval \[0, 3\]. We have notable points like: \ f(0) = 0 \, \ f(1) = -2 \, and \ f(3) \ (which can be calculated using cube roots and exponentiation).
• Look for specific patterns such as increasing, decreasing, and inflection points.
• To get precise values, calculating at non-integer points might be beneficial.

Sketching becomes easier when relying on critical points and understanding the function's overall behavior. Analyze the curvature based on these key points.

Continuity

For any function to apply Rolle's Theorem, it must be continuous over the closed interval in question. In this exercise: \[ f(x) = x^{4/3} - 3x^{1/3} \] is composed of two basic continuous functions:

• \[ x^{4/3} \]
• \[ -3x^{1/3} \]

When combined, their sum remains continuous over the interval \[0, 3\]. A function is continuous if it has no breaks, holes, or jumps in the domain. Thus, it's crucial to ascertain this property before moving ahead with Rolle's Theorem.

Differentiability

Another key criterion for Rolle's Theorem is differentiability over an open interval. Here, for \[ f(x) = x^{4/3} - 3x^{1/3} \] let's find the derivative: \[ f'(x) = \frac{4}{3} x^{1/3} - \frac{1}{x^{2/3}} \] This derivative tells us the slope of the tangent line to the curve at any point. For differentiability:

• The function must have a defined derivative at every point in the open interval.
• \[ f'(x) \] shows a potential issue at \[x = 0\] because the term \[ \frac{1}{ x^{2/3} }\] becomes undefined.

However, at any point away from \[0\] within (0, 3), the function is differentiable.

Horizontal Tangent Line

To apply Rolle's Theorem in finding where a horizontal tangent line exists, we look for points where \[ f'(x)=0 \]. From the derivative: \[ f'(x) = \frac{4}{3} x^{1/3} - \frac{1}{x^{2/3}} \] Setting this to zero: \[ \frac{4}{3} x^{1/3} - \frac{1}{x^{2/3}} = 0 \] Our task is to solve this: \[ \frac{4}{3} x = 1 \] or, \[ x = (\frac{3}{4})^{3/2} \].
This result points us to where the slope is zero, representing the existence of a horizontal tangent line. This tangency implies the function’s rate of change is momentarily zero, a key insight derived from Rolle’s Theorem.