Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line is horizontal.

Verified step by step guidance

1. Step 1: Understand that a horizontal tangent line occurs where the derivative of the function is zero. Therefore, we need to find the derivative of the function f(x) = 4√x - x.
2. Step 2: Differentiate f(x) with respect to x. The derivative of 4√x is 4 * (1/2)x^(-1/2) = 2x^(-1/2), and the derivative of -x is -1. So, f'(x) = 2x^(-1/2) - 1.
3. Step 3: Set the derivative equal to zero to find the x-values where the tangent line is horizontal: 2x^(-1/2) - 1 = 0.
4. Step 4: Solve the equation 2x^(-1/2) - 1 = 0 for x. This involves isolating x by first adding 1 to both sides, then multiplying both sides by x^(1/2), and finally squaring both sides to solve for x.
5. Step 5: Once you have the x-value(s), substitute back into the original function f(x) to find the corresponding y-value(s). These (x, y) pairs are the points on the graph where the tangent line is horizontal.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the graph of the function at any given point. To find points where the tangent line is horizontal, we need to set the derivative equal to zero.

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are significant because they can indicate local maxima, minima, or points of inflection. In the context of finding horizontal tangents, we focus on points where the derivative equals zero, as these correspond to horizontal tangent lines on the graph.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is given by the derivative of the function at that point. A horizontal tangent line has a slope of zero, which means we are looking for points where the derivative of the function equals zero.