Suppose f has a continuous derivative...?

Question

Suppose f has a continuous derivative whose values are given in the following table.

a) Estimate the x-coordinates of critical points of f for 0(less than or equal to) x(less than or equal to)10.

b) For each critical point, indicate if it is a local maximum of f, local minimum, or neither.

x: 0, 1 , 2, 3, 4, 5, 6, 7, 8, 9, 10
f'(x): 5, 2, 1, -2, -5, -3, -1, 2, 3, 1, -1

Answer 1

A critical point is where the derivative is 0.

If a function F is continuous, and F(A) < 0 and F(B) > 0, then there is a point C, between A and B where F(C) is 0.

In your case, the function whose roots (the points where it is 0) that you are looking for is the derivative, f'. f'(2) = 1 and f'(3) = -2 so there has to be at least one critical point in there. Similarly between x = 6 and x = 7, and between x = 9 and x = 10.

If you have the function at two points and want to find the value in between, you have to interpolate. The simplest interpolation method is linear interpolation: drawing a line between the two point on either side of the region.

Consider F(1) = 1 and F(2) = -2. The slope of the line is

m = (F(2) - F(1))/(2-1) = -3

So the equation of the line is:

y = m(x-1) + 1

solve for y = 0 to get an approximation to the first inflection point. Similarly for the others.

A critical point is a local maximum if the function decreases with increasing x. That happens if the derivative is negative after the critical point. Vice versa for local minima.