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 ___?

Length of normal = y√(1 + (dy/dx)²)

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 ___?

Tan(θ) = |(f'(x1) - g'(x1)) / (1 + f'(x1)g'(x1))|

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

F'(x) ≥ 0 for all x in (a, b).

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 or at the endpoints a and 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 ___?

F'(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 = y (dy/dx).

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.