Why is the Second Derivative Positive if a Function is Concave Up in Mathematics?

If f(x) is concave up, then f”(x) is?

If a function f(x) is concave up, then its second derivative, f”(x), is positive

If a function f(x) is concave up, then its second derivative, f”(x), is positive.

To understand why this is the case, let’s recall the definition of concavity. A function is concave up if the curve lies above its tangent line at each point. In other words, the function is curving upward.

The second derivative f”(x) represents the rate of change of the first derivative f'(x). If f”(x) is positive, it means that the slope of the tangent line (represented by f'(x)) is increasing. In other words, as x increases, the slopes of the tangent lines are getting steeper and the function is curving more and more upward.

On the other hand, if f”(x) were negative, it would imply that the slope of the tangent lines (f'(x)) is decreasing. In this case, the function would be curving downward or concave down, instead of being concave up.

In summary, if a function f(x) is concave up, its second derivative f”(x) will be positive.

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