Question

Find the \(x\) -values where the graph of the function has a horizontal tangent line. $$f(x)=6 x^{2}-18 x-24$$

Short answer

The graph has a horizontal tangent line at \(x = \frac{3}{2}\).

Step-by-step solution

Understand Horizontal Tangent Line

A horizontal tangent line on the graph of a function occurs where the derivative of the function is zero. Therefore, we need to find the derivative of the given function and solve for where it equals zero.

Find the Derivative of the Function

The function given is: \[ f(x) = 6x^2 - 18x - 24 \]Use the power rule, which states that the derivative of \(ax^n\) is \(nax^{n-1}\), to find the derivative of the function:\[ f'(x) = \frac{d}{dx}(6x^2) - \frac{d}{dx}(18x) - \frac{d}{dx}(24) \]\[ f'(x) = 2 \cdot 6x^{2-1} - 18 = 12x - 18 \]

Set the Derivative Equal to Zero

To find where the graph has a horizontal tangent line, set the derivative equal to zero and solve for \(x\): \[ 12x - 18 = 0 \]

Solve for \(x\)

Solve the equation from Step 3 for \(x\):\[ 12x - 18 = 0 \]Add 18 to both sides: \[ 12x = 18 \]Divide by 12:\[ x = \frac{18}{12} \]Simplify the fraction:\[ x = \frac{3}{2} \]

Key concepts

Horizontal Tangent Line

A horizontal tangent line on the graph of a function is a straight line that barely touches the curve at a point. But, it's not slanted; hence, it's horizontal. This occurs when the slope of the tangent line is zero. Since the slope of a tangent line at any point on a function is represented by the derivative of that function, a horizontal tangent line appears wherever the derivative equals zero.

Think of it this way: the derivative tells you how the function is climbing or falling — its slope. When this slope is zero, the function flattens momentarily, creating a horizontal tangent. Therefore, to find where horizontal tangent lines appear in a function, one must simply set the derivative of the function equal to zero and solve for the variable, often denoted as \( x \).

Power Rule

The power rule is a shortcut in calculus used for finding the derivative of a power function. It's incredibly useful and simplifies what would otherwise be arduous calculations. The power rule states: if you have a function of the form \( ax^n \), its derivative is \( n \cdot ax^{n-1} \). This means you bring the exponent down as a multiplier and subtract one from the original exponent.

For example, when applying the power rule to the function \( f(x) = 6x^2 - 18x - 24 \), the derivative becomes \( 12x - 18 \). This is calculated by first finding the derivative of each term separately and then adding them:

• \( 6x^2 \) becomes \( 12x \)
• \( -18x \) becomes \( -18 \)
• The constant \( -24 \) becomes 0 (since the derivative of a constant is always 0)

Each term's derivative is added together to get the final derivative of the whole function.

Graph of a Function

The graph of a function is a visual representation of all the points that satisfy a given function \( f(x) \) in a coordinate system. It shows you how the function behaves for different values of \( x \).

When we talk about the graph of the function \( f(x) = 6x^2 - 18x - 24 \), it’s a parabola because its highest degree term is \( x^2 \). This quadratic function graph opens upwards, having a specific vertex position depending on its coefficients. Observing the graph can help identify minimums, maximums, or flat sections where horizontal tangents could occur.

Such visual cues facilitate understanding where function behavior potentially changes and offer insight when solving problems involving derivatives or tangent lines.

Solve for x

Solving for \( x \) involves finding the solutions or the values that make an equation true. In the context of finding where a graph has a horizontal tangent line, solving for \( x \) means finding the points where the derivative equals zero.

From the derivative \( 12x - 18 = 0 \), we solve step by step:

• Add 18 to both sides to isolate terms involving \( x \): \( 12x = 18 \)
• To solve for \( x \), divide both sides by 12: \( x = \frac{18}{12} \)
• Simplify the fraction: \( x = \frac{3}{2} \)

The solution, \( x = \frac{3}{2} \), indicates the \( x \)-value where the function’s graph has a horizontal tangent line. This simple equation-operating technique is critical for determining key points in calculus problems.