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Extreme Value Theorem

If a function f(x) is continuous on a closed interval, the f(x) has both a maximum and a minimum value in the interval.

critical point

a point in the domain of the function at which the derivative is zero or undefined.

Rolle's Theorem

If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists a point c∈(a,b), where f '(c) = 0.

Mean Value Theorem

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b)

It is increasing

If f '(x) > 0 on (a,b), then f(x) has what characteristic?

It is decreasing

If f '(x) < 0 on (a,b), then f(x) has what characteristic?

It is constant

If f '(x) = 0 on (a,b), then f(x) has what characteristic?

To find a relative maximum on f(x) (by the 1st derivative test)...

Find a point where f '(x) is zero or undefined, AND where f '(x) changes from positive to negative.

To find a relative minimum on f(x) (by the 1st derivative test)...

Find a point where f '(x) is zero or undefined, AND where f '(x) changes from negative to positive.

It is concave upward

If f "(x) > 0 on (a,b), then f(x) has what characteristic?

It is concave downward

If f "(x) < 0 on (a,b), then f(x) has what characteristic?

It is linear

If f "(x) = 0 on (a,b), then f(x) has what characteristic?

Inflection Point

a point in the domain of the function at which a tangent line exists and the concavity of the function changes.

To find a point of inflection on f(x)...

Find a point where f "(x) is zero or undefined, where a tangent line exists, AND where f "(x) changes sign.

To find a relative maximum on f(x) (by the 2nd derivative test)...

Find a point where f '(x) is zero AND where f "(x) is negative.

To find a relative minimum on f(x) (by the 2nd derivative test)...

Find a point where f '(x) is zero AND where f "(x) is positive.

f '(x) * dx

If y = f(x) is a differentiable function and dx ≠ 0, then dy =

Tangent Line

Another name for a linear approximation to the curve is a

f '(a)*(x − a) + f(a)

The linear approximation of f(x) near x = a is