Question

Find the exact x-coordinate where there is a horizontal tangent line to the graph of f(x)=x10 x

. Type ln(x) for the natural logarithm function. 1)b) Let f(x)=(lnx) 3
f ′
(x)=
​
f ′
(e 3
)= 1)c) Let f(x)=−5log 5
​
(x)

Answer 1

In mathematics, to find the x-coordinate where a function has a horizontal tangent line, you calculate the derivative of the function, set it equal to zero, and solve for x. This indicates the points where the slope of the tangent line is zero, which is the case for a horizontal line. The concepts of natural logarithm (ln) and common logarithm (log) often play a role in these calculations.

Step-by-step explanation:

To find the x-coordinate where there is a horizontal tangent line to the graph of a function, you need to find where the derivative of the function equals zero. This is because the derivative represents the slope of the tangent line to the graph at any given point. For a horizontal tangent line, the slope must be zero.

For example, consider the function f(x) = (ln(x))^3. To find where the function has a horizontal tangent line, we first find its derivative, which is f'(x) = 3(ln(x))^2 * (1/x). Setting the derivative equal to zero, f'(x) = 0, and solving for x gives us the x-coordinate of the points where there are horizontal tangent lines.

Similarly, for a function such as f(x) = -5log5(x), we would find the derivative and then set it equal to zero to find the x-coordinate of the horizontal tangent line.

The concept of natural logarithm (ln) and the common logarithm (log) is essential in these types of problems. The natural logarithm of a number (ln) is the power to which e must be raised to equal the number, where e is the mathematical constant approximately equal to 2.7182818. The common logarithm of a number is the power to which 10 must be raised to equal the number.