What does the negative slope mean?

Visually, a line has negative slope if it goes down and right (or up and left). Mathematically, this means that as x increases, y decreases.

How do you know if a critical point is maximum or minimum?

Determine whether each of these critical points is the location of a maximum, minimum, or point of inflection. For each value, test an x-value slightly smaller and slightly larger than that x-value. If both are smaller than f(x), then it is a maximum. If both are larger than f(x), then it is a minimum.

How do you know if a critical point is max or min using the second derivative?

If a function has a critical point for which f′(x) = 0 and the second derivative is positive at this point, then f has a local minimum here. If, however, the function has a critical point for which f′(x) = 0 and the second derivative is negative at this point, then f has local maximum here.

Can a critical point be negative?

The second derivative test: If f ”(x) exists at x0 and is positive, then f ”(x) is concave up at x0. If f ”(x0) exists and is negative, then f(x) is concave down at x0. Definition of a critical point: a critical point on f(x) occurs at x0 if and only if either f ‘(x0) is zero or the derivative doesn’t exist.

What to do if the slope is negative?

If the slope is negative, then the rise and the run have to be opposites of each other, one has to be positive and one has to be negative. In other words, you will be going up and to the left OR down and to the right.

What can be said regarding of a line if its slope is negative?

Hence, we can say that the line with negative slope makes an obtuse angle with the x-axis when measured anti-clockwise.

How do you prove a point is maximum?

Method 1: If f'(x)>0 for all a and if f'(x)<0 for all c0 for all c

What does it mean if the second derivative is less than 0?

concave down
The second derivative of f(x) tells us the rate of change of the derivative f (x) of f(x). The second derivative is negative (f (x) < 0): When the second derivative is negative, the function f(x) is concave down.

Is negative to positive a max or min?

Simply, if the first derivative is negative to the left of the critical point, and positive to the right of it, it is a relative minimum. If the first derivative test finds the first derivative is positive to the left of the critical point, and negative to the right of it, the critical point is a relative maximum.

Can the local maximum be negative?

If the derivative is negative to the right of a left endpoint, the function has a local maximum at the left endpoint. The plus and minus signs indicate the sign of the first derivative, which also represents the slope of the tangent lines and indicate whether the function is increasing or decreasing.

What happens if the second derivative of a graph is negative?

Similarly if the second derivative is negative, the graph is concave down. This is of particular interest at a critical point where the tangent line is flat and concavity tells us if we have a relative minimum or maximum. Second derivative test of extrema: Let f ( x) be a function with . f ′ ( x 0) = 0.

What is the second derivative test for critical points?

The point x may be a local maximum or a local minimum, and the function may also be increasing or decreasing at that point. The three cases above, when the second derivative is positive, negative, or zero, are collectively called the second derivative test for critical points.

What is the second derivative of a function with a slope?

When a function’s slope is zero at x, and the second derivative at x is: less than 0, it is a local maximum. greater than 0, it is a local minimum. equal to 0, then the test fails (there may be other ways of finding out though) “Second Derivative: less than 0 is a maximum, greater than 0 is a minimum”.

What happens when the slope of a function is zero?

When a function’s slope is zero at x, and the second derivative at x is: 1 less than 0, it is a local maximum 2 greater than 0, it is a local minimum 3 equal to 0, then the test fails (there may be other ways of finding out though)