Understanding Concave Down: Role of the Second Derivative in Calculus

If f(x) is concave down, then f”(x) is?

If a function f(x) is concave down, it means that its graph has a downward curvature, resembling the shape of a cup or a frown

If a function f(x) is concave down, it means that its graph has a downward curvature, resembling the shape of a cup or a frown. In terms of calculus, this implies that the second derivative of f(x), denoted as f”(x), is negative.

To understand this relationship, let’s go through the definition of concavity and discuss the role of the second derivative.

Concavity:

A function f(x) is concave down on an interval if, for any two points in that interval (let’s call them x1 and x2), the value of f(x1) is greater than the value of f(x2), and the graph of f(x) lies below the line segment connecting the two points.

Mathematically, if f(x) is concave down, then it satisfies the inequality:

f(tx1 + (1-t)x2) ≤ tf(x1) + (1-t)f(x2),

for any value of t between 0 and 1, and for any x1 and x2 in the interval.

Second Derivative:

The second derivative measures the concavity of a function. If f”(x) > 0, it implies that the graph of f(x) is concave up, resembling the shape of a smiley face. On the other hand, if f”(x) < 0, it indicates that the graph of f(x) is concave down, forming a frown or a cup shape. To see why this is the case, consider two points x1 and x2 in the interval where f(x) is concave down. The slope of the tangent line at each of these points is given by the first derivative f'(x). If f''(x) < 0, it means that f'(x) is decreasing as x moves from x1 to x2. This decrease in slope corresponds to the downward curvature of the graph, which satisfies the definition of concavity. Hence, when f(x) is concave down, we can conclude that f''(x) < 0. Note: It's important to remember that this relationship holds for functions that are twice differentiable and have a continuous second derivative. In some rare cases, a function may not have a second derivative at certain points or may have points of inflection where concavity changes.

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