If f^''(x)<0 ∀x ∈(a, b), then f^'(x)=0 (A) exactly once in (a, b) (B) at most once in (a, b) (C)

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If f^''(x)<0 ∀x ∈(a, b), then f^'(x)=0 (A) exactly once in...

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Key Concepts

Concavity

Concavity is a property of functions determined by the sign of the second derivative. If the second derivative is strictly negative on an interval, the function is strictly concave down, meaning that the slope of the function is decreasing. This characteristic influences many properties of the function, including the behavior of its first derivative and the nature of its critical points.

Monotonicity of the First Derivative

When the second derivative is strictly negative, it implies that the first derivative is strictly decreasing over the interval. A strictly decreasing function will cross a particular value, such as zero, at most once. This is a direct consequence of the monotonicity ensuring that the function cannot oscillate or have repeated values.

Critical Points in Strictly Concave Functions

A critical point is where the first derivative is zero, which often corresponds to a local maximum or minimum. In the case of a strictly concave function, if a critical point exists, it is unique because the first derivative, being strictly decreasing, can only change sign once. This unique behavior helps in determining whether the function attains a unique local extremum.

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