Problem 14 - 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. $$ f(x)=x^{3 / 4}-2 x^{1 / 4} ;[0,4] $$

Step-by-Step Solution

Answer

Condition (iii) of Rolle's theorem fails because \( f(0) eq f(4) \). Hence, there's no guarantee of a horizontal tangent line in \( [0, 4] \).

Step 1 - Sketch the Graph

To sketch the graph of the function \( f(x) = x^{3/4} - 2x^{1/4} \) on the interval [0, 4], plot several key points. Evaluate the function at a few selected points within the interval to understand its shape. For instance, compute \( f(0) \), \( f(1) \), \( f(2) \), and \( f(4) \). Then connect these points smoothly to approximate the graph: - \( f(0) = (0)^{3/4} - 2(0)^{1/4} = 0 \) - \( f(1) = (1)^{3/4} - 2(1)^{1/4} = 1 - 2 = -1 \) - \( f(2) = (2)^{3/4} - 2(2)^{1/4} \, \approx 1.32 - 2(1.19) \, \approx 1.32 - 2.38 = -1.06 \) - \( f(4) = (4)^{3/4} - 2(4)^{1/4} \approx 2.52 - 2(1.41) \approx 2.52 - 2.82 = -0.30 \)

Step 2 - Verify Condition (i) of Rolle's Theorem

Condition (i) states that the function must be continuous on the closed interval [0, 4]. Since \( f(x) \) is composed of polynomial and root functions, which are continuous wherever they are defined, \( f(x) = x^{3/4} - 2x^{1/4} \) is continuous on [0, 4].

Step 3 - Verify Condition (ii) of Rolle's Theorem

Condition (ii) requires the function to be differentiable on the open interval (0, 4). Differentiability implies continuity, and the function involving \( x^{3/4} \) and \( x^{1/4} \) is differentiable where \( x > 0 \). Hence, \( f(x) \) is differentiable on (0, 4).

Step 4 - Verify Condition (iii) of Rolle's Theorem

Condition (iii) states that the values of the function at the endpoints of the interval must be equal, i.e., \( f(0) = f(4) \). From the previous step, \( f(0) = 0 \) and \( f(4) \approx -0.30 eq 0 \). Since \( f(0) eq f(4) \), Condition (iii) is not satisfied.

Step 5 - Conclusion on Horizontal Tangent Line

Since all three conditions of Rolle's Theorem are not satisfied, specifically Condition (iii), there is no guarantee of a horizontal tangent line for the function on the interval [0, 4].

Key Concepts

Graph sketching

Graph sketching is an essential skill in calculus that helps you visualize the behavior of functions. For the function \text{ \( f(x) = x^{3/4} - 2x^{1/4} \)},s sketching its graph involves evaluating the function at a few key points within the interval [0, 4]. This gives you an idea of its shape. Plotting:

• \( f(0) = 0 \)
• \( f(1) = -1 \)
• \( f(2) \approx -1.06 \)
• \( f(4) \approx -0.30 \)

helps in approximating the function's curve within this interval. By connecting these points smoothly, you can sketch a graph that shows the function’s behavior.

Continuity

Continuity means a function has no breaks, jumps, or holes within a specific interval. To satisfy Rolle's Theorem, the function must be continuous on a closed interval, in this case, [0, 4]. The function \( f(x) = x^{3/4} - 2x^{1/4} \) is composed of terms with polynomial and root functions, which are continuous where they are defined. Since both components are continuous for all values of x in [0, 4], the function l\text{\( f(x) = x^{3/4} - 2x^{1/4} \) } is continuous on [0, 4].

Differentiability

Differentiability refers to a function's ability to have a derivative at every point in its domain. For Rolle's Theorem, the function must be differentiable on the open interval \text{(0, 4)} . Differentiable functions are always continuous, but the converse is not always true. Considering \text{\( f(x) = x^{3/4} - 2x^{1/4} \) is } differentiable where \text{x} is greater than zero (0). So \( f(x) = x^{3/4} - 2x^{1/4} \) is differentiable in the open interval (0, 4) since it’s possible to take derivatives of both terms: \text{\( f'(x) = \frac{3}{4} x^{-1/4} - \frac{2}{4} x^{-3/4} \)}

Horizontal tangent line

A horizontal tangent line occurs where the derivative of the function equals zero. For Rolle's Theorem to predict the existence of these lines, the function must satisfy three conditions: continuity on [0, 4], differentiability on (0, 4), and the endpoints of the interval must yield equal function values \text{\( f(0) = f(4) \) } as seen before,\ \text{ \( f(0) = 0 \) } while\text{\( f(4) \approx -0.30 \) } Since 0 does not equal -0.30, condition (iii) is violated. Consequently, we cannot guarantee a point with a horizontal tangent line in the interval [0, 4].