Question

Tangent lines and concavity Give an argument to support the claim that if a function is concave up at a point, then the tangent line at that point lies below the curve near that point.

Short answer

Question: Provide an explanation supporting the claim that if a function is concave up at a point, then the tangent line at that point lies below the curve near that point. Answer: A function is concave up at a point if its second derivative is positive at that point, causing the curve to bend upward. The tangent line at that point has a slope equal to the function's first derivative. Since the second derivative is positive, the first derivative is increasing near the point, meaning the tangent line's slope is less than the slopes of the curve to the right of the point, causing the curve to be above the tangent line. Similarly, to the left of the point, the tangent line's slope is greater, so the curve remains below the tangent line. Thus, if a function is concave up at a point, the tangent line lies below the curve near that point.

Step-by-step solution

Definition of Concave Up

Recall the definition of a function being concave up at a point. A function f(x) is concave up at a point x=a if the function's second derivative, f''(a), is positive, i.e., f''(a) > 0. In this scenario, the function's curve is bending upward at x=a.

Relation Between Tangent Line and Function Curve

The tangent line at a point on the function curve is the best linear approximation of the function near that point. The tangent line's slope is equal to the function's first derivative, f'(a), at the point x=a.

Establish Direct Relation Between f'(x) and f''(x)

When f''(a) > 0, it means that the first derivative of the function, f'(x), is increasing as x goes from left to right about the point x=a. In other words, the tangent line's slope is increasing near the point x=a.

Argument for Tangent Line Being Below the Curve for Concave Up Functions

As the tangent line at x=a has a slope of f'(a), and f'(x) is increasing in this interval, it means that the tangent line's slope is less than the slopes of the curve to the right of the point x=a. Since the curve has a greater slope than the tangent line to its right, it means that the curve is growing faster than the tangent line; therefore, the curve will be above the tangent line to the right of x=a. Similarly, to the left of the point x=a, the tangent line's slope is greater than the slope of the curve, meaning the tangent line grows faster than the curve, and hence, the curve remains below the tangent line. Thus, when a function is concave up at a point, the tangent line lies below the curve near that point.