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- Continuous on the closed interval [a,b]

What are the requirements for the Mean Value Theorem?

What does the Mean Value Theorem guarantee?

That there is a C in the interval (a,b) such that f'(c)= f(b) - f(a)/ b - a

The relative minimums and maximums

What are relative extrema of f(x)?

When do critical values of f(x) occur?

When the derivative of f(x) is zero or undefined

When f(x) goes from increasing to decreasing

With Respect to f(x), when do relative maximums occur?

When f'(x) goes from positive to negative

With respect to the derivative of f(x), when do relative maximums occur?

when f(x) goes from decreasing to increasing

With respect to f(x), when do relative minimums occur?

When f'(x) goes from negative to positive

With respect to the derivative of f(x), when do relative minimums occur?

when f'(x) is positive

With respect to its derivative, when is f(x) increasing?

When f'(x) is negative

With respect to its derivative, when is f(x) decreasing?

f''(x) needs to change signs there

What are the two requirements for a point of inflection on f(x) to occur?

when f'(x) is increasing

In regards to f'(x), when is f(x) concave up?

when f''(x) is positive

In regards to f''(x), when is f(x) concave up?

When f'(x) is decreasing

In regards to f'(x), when is f(x) concave down?

When f"(x) is negative

In regards to f"(x), when is f(x) concave down?

6

Find c that satisfies the MVT: f(x) = 5x^2 - 4x - 10 [4,8]

π/2 and 3π/2

Find c that satisfies the MVT: f(x) = sinx [0,2π]

MVT doesn't hold true

Find c that satisfies the MVT f(x) = 1/(x + 3) [-5, 0]

How do you find absolute extrema?

Find the critical points, make a number line, determine the max or min inside the interval, then consider the endpoints and if they are max or min. Finally, make and x, y chart and compare the y-values

What is an absolute maximum?

The biggest y-value on the given interval

What does the Intermediate Value Theorem guarantee?

That there is a y-value between f(a) and f(b) where the corresponding x-value is between a and b

f(x) must be continuous on [a,b]

What is the restriction of the IVT?