First Derivative graphs

f(x) has a derivative of zero when the graph of f has a relative min or max.

If the derivative of f(x) changes from positive to negative at some point, then the graph of f has a relative minimum at that point.

If the derivative of f(x) is always positive, then the graph of f has no relative extrema.

If the graph of f is always increasing, then the derivative of f(x) is never negative.

(A)

(B)

(C)

(D)

A.

B.

C.

D.

x=-5 and -1

x=-3

x=-1

x=-5

$$f\left(x\right)$$  is increasing

$$f\left(x\right)$$  is decreasing

$$f\left(x\right)$$  is positive

$$f\left(x\right)$$  is negative

local maximum

local minimum

stationary point/platuae point

local maximum

local minimum

stationary point

When a graph is increasing, its derivative is negative.

When a graph is decreasing, so is its derivative.

When a graph is decreasing, its derivative is negative.

When a graph is increasing, so is its derivative.

x=-5 and -1

x=-3

x=-1

x=-5

x=-5 and -1

x=-3

x=-1

x=-5

At x=-2 f(x) is increasing because the graph has a positive slope.

At x=-2 f(x) is decreasing because the graph has a positive slope.

At x=-2 f(x) is decreasing because the graph lies below the x-axis.

At x=-2 f(x) is decreasing because the graph has a negative slope

A

B

C

D