Horizontal Tangent Line

Question

Determine the points at which the graph of the function has a horizontal tangent line.

y = 3x + 2 cos(x),    0 ≤ x < 2𝜋

STEP 1:

Find the derivative.

y ′ = 

STEP 2:

Set y ′ = 0 and solve for x.

smaller x-valuex_{1 =}

larger x-value x_{2 =}

STEP 3:

Find the y values by substituting the values from Step 2 into the original function. y₁ corresponds to x₁ and y₂ corresponds to x₂. List the points where the function has horizontal tangent lines.

Answer 1

This is exactly a "pretty" question, so your answers will probably include some rounded decimal values. Take a look at the steps given as the three main parts of the process to find the points where a function has a horizontal tangent line.

Step 1: To find the derivative of y = 3x + 2 cos(x), you'll need to use some chain rule for the square root of 3x and then the derivative of cosine. When simplified a little, you'll have a fraction as your first term, and a negative second term.

Step 2: Write down your derivative and put it equal to 0. Now when the derivative is something "nice" like a linear or quadratic equation, you can solve for x with a little algebra. But the derivative here will have a fraction and a trig function in it, so it isn't really "nice". This means you have to solve for x another way - probably with a graphing calculator or other graphing software. You will want to look for the x-values where the graph of the derivative crosses or touches the x-axis. It's also usually good to note if there is any x-value where the graph is undefined. The problem tells you to only use the function for 0 ≤ x < 2𝜋, so focus your search between x = 0 and x = 6.28 roughly. There should be two x-values where the derivative crosses the x-axis, since you are asked for two answers. You will probably need to round the values, pay attention to if the problem tells you how many decimal places.

Step 3: Now take your answers from Step 2, and plug them into the original equation y = 3x + 2 cos(x) as inputs for x. Use a calculator to find the output or y-value for each of the two x-values from Step 2. Again, you will probably have to round some decimals, and make sure your calculator is in radian mode. Once you have the y-values, write your final answers as points (x1, y1) and (x2, y2). These will be the points on the graph of y = 3x + 2 cos(x) where you could draw a horizontal line and it would look like it just barely grazes the graph. Or in other words, they are the points on the graph where for a brief moment the slope of the graph is zero.