If f(x) has a local maximum at x = a, then f'(a) must be ___?

0

True or False: The second derivative test for concavity states that if f''(x) > 0, the function is concave downward.

False. If f''(x) > 0, the function is concave upward.

The equation of the tangent line to the curve y = f(x) at point P(x₁,y₁) is given by ___?

Y - y₁ = f'(x₁)(x - x₁)

The length of the normal at point P(x, y) on the curve y = f(x) is given by ___?

If two curves y = f(x) and y = g(x) intersect at point P, the angle between the two curves can be found using the formula ___?

A function f(x) is said to be monotonic increasing on the interval [a, b] if ___?

The necessary condition for the existence of an extremum of function f(x) is that ___?

F'(x) = 0 or f'(x) does not exist at x.

The first derivative test states that if f'(x) changes from negative to positive at x = c, then f(c) is a ___?

Local minimum.

True or False: If f''(c) > 0, then the function has a local maximum at x = c.

False. If f''(c) > 0, the function has a local minimum at x = c.

The maximum and minimum values of a continuous function on [a, b] can occur at ___?

Critical points within (a,b) or at the endpoints x = a or x = b.

If y = f(x) is represented parametrically as x = g(t) and y = h(t), the condition for finding extrema is ___?

h'(t) = 0 and g'(t) ≠ 0.

The tangent to the curve at point P(x₁, y₁) has the slope given by ___?

slope of the tangent = f′(x₁)

where f′(x₁) is the derivative of the function y = f(x) evaluated at x = x₁.

The equation for the length of the subnormal at point P(x, y) on the curve y = f(x) is given by ___?

Length of subnormal =

If f'(x) = 0 for all x in an interval I, then f(x) is ___?

Constant on that interval.

The derivative of a function f(x) at a point is defined as the ___ of the function at that point.

Slope of the tangent line to the curve y = f(x) at that point.