Problem 70 - Calculus for Biology and Medicine

Question

Find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=3 x^{5}-\frac{3}{2} x^{4} $$

Step-by-Step Solution

Answer

The points with horizontal tangents are (0, 0) and \(\left(\frac{2}{5}, \frac{48}{3125}\right)\).

Step 1: Differentiate the Function

To find the horizontal tangents, we start by finding the derivative of the function, since horizontal tangents occur where the derivative equals zero. Differentiate the given function: \[ f(x) = 3x^5 - \frac{3}{2}x^4 \]Apply the power rule to each term to get the derivative:\[ f'(x) = 15x^4 - 6x^3 \]

Step 2: Set the Derivative Equal to Zero

Set the derivative found in Step 1 equal to zero to find the critical points where horizontal tangents occur:\[ 15x^4 - 6x^3 = 0 \]

Step 3: Factor the Equation

The equation from Step 2 can be factored to solve for the values of \(x\):\[ 3x^3(5x - 2) = 0 \]This gives two factors: \(3x^3 = 0\) and \(5x - 2 = 0\).

Step 4: Solve Each Factor

Solve the first factor:\[ 3x^3 = 0 \Rightarrow x = 0 \]Solve the second factor:\[ 5x - 2 = 0 \Rightarrow 5x = 2 \Rightarrow x = \frac{2}{5} \]

Step 5: Evaluate the Function at Critical Points

Substitute the critical points back into the original function to find the \(y\)-coordinates:For \(x = 0\):\[ f(0) = 3(0)^5 - \frac{3}{2}(0)^4 = 0 \]For \(x = \frac{2}{5}\):\[ f\left(\frac{2}{5}\right) = 3\left(\frac{2}{5}\right)^5 - \frac{3}{2}\left(\frac{2}{5}\right)^4 \]Simplify,\[ f\left(\frac{2}{5}\right) = 3 \cdot \frac{32}{3125} - \frac{3}{2} \cdot \frac{16}{625} = \frac{96}{3125} - \frac{48}{3125} = \frac{48}{3125} \]

Step 6: Write Down the Points with Horizontal Tangents

Using the solutions from Step 5, the points on the graph of \(y=f(x)\) with horizontal tangents are:- \((0, 0)\)- \(\left(\frac{2}{5}, \frac{48}{3125}\right)\)

Key Concepts

Derivative

In calculus, the derivative is a fundamental concept that allows us to understand how a function changes at any given point. When you differentiate a function, you essentially find another function which gives the slope of the original function at each point.
The process of differentiation can help determine:

• The rate at which quantities change.
• The slope of the tangent to a curve at any point.

To find the derivative of a polynomial function like \( f(x) = 3x^5 - \frac{3}{2}x^4 \), we apply the power rule to each term to result in a simpler function, \( f'(x) = 15x^4 - 6x^3 \). This new function represents the rate of change of \( f(x) \), and its value at any point gives the slope of the tangent at that point.

Horizontal Tangent

A horizontal tangent line on a graph indicates where the function has a zero slope. That means the derivative at those points is zero.
In context, when you set the derivative of the function equal to zero \( f'(x) = 0 \), and solve for \( x \), you find critical points where potentially horizontal tangents occur. These aren't necessarily maximums or minimums in every scenario, but they do show where the function stops increasing or decreasing momentarily. This is key in identifying the behavior of graphs and identifying specific types of critical points.

Critical Points

Critical points of a function are where the derivative is zero or undefined. To find these, you solve \( f'(x) = 0 \).
After finding the derivative of our function \( f(x) \), which is \( 15x^4 - 6x^3 \), we set it to zero to determine our critical points. In our solution, we found:

• \( x = 0 \)
• \( x = \frac{2}{5} \)

These values of \( x \) indicate potentially interesting points on the graph where the slope is zero, and, therefore, possibly horizontal tangents. Evaluating these in the original function gives us the full coordinates of these critical points.

Power Rule

The power rule is a straightforward differentiation technique used specifically for polynomial terms. It states that for any term \( ax^n \), its derivative is \( anx^{n-1} \). The power rule is instrumental for breaking down polynomials into simpler derivatives.
For example, applying the power rule to each term in \( f(x) = 3x^5 - \frac{3}{2}x^4 \) gives us \( f'(x) = 15x^4 - 6x^3 \). This step is often the first in identifying critical points, points of inflection, or examining the behavior of a function's graph in more detail.