If a function f(x) is concave at x = a, then f”(a) < 0.
If a function f(x) is concave at x = a, then f”(a) < 0.
The correct option is A True
In the interval in which a function is concave, the value of the derivative f’(x) keeps on reducing. So we can say f”(x) will be less than zero.
One example would be y = - x2
The shape of this function is downward parabola. This is a typical example of a concave function. Here, as you can see from the graph, the slope of tangent reduces as x increases. Initially the slope of the tangent (f’(x) ) is positive when x is negative and slope becomes zero at x =0. For positive values of x, the slope is is negative. So we can say the slope or f’(x) decreases as x increases. So the derivative of f’(x) should be negative.
Any point on the x-axis would have f”(x) negative in case of concave function - x2
True
In the interval in which a function is concave, the value of the derivative f’(x) keeps on reducing. So we can say f”(x) will be less than zero.
One example would be y = - x2
The shape of this function is downward parabola. This is a typical example of a concave function. Here, as you can see from the graph, the slope of tangent reduces as x increases. Initially the slope of the tangent (f’(x) ) is positive when x is negative and slope becomes zero at x =0. For positive values of x, the slope is is negative. So we can say the slope or f’(x) decreases as x increases. So the derivative of f’(x) should be negative.
Any point on the x-axis would have f”(x) negative in case of concave function - x2
