Connecting f(x), f'(x), & f''(x)

Derivative means the same thing as

• slope of the tangent line

• slope of the normal line

• slope of the secant line

• the function’s value

If f(x) is DECREASING and CONCAVE UP...

• f'(x) is positive & & f"(x) is positive

• f'(x) is negative & f"(x) is positive

• f'(x) positive & f"(x) is negative

• f'(x) negative & f"(x) is negative

Tags

CCSS.HSF.IF.B.4

If f'(x) is DECREASING...

• f(x) is concave up, f"(x) is positive

• f(x) is concave down, f"(x) is positive

• f(x) is concave up, f"(x) is negative

• f(x) is concave down, f"(x) is negative

If f"(x) is NEGATIVE

• f(x) is concave up & f'(x) is increasing

• f(x) is concave up & f'(x) is decreasing

• f(x) is concave down & f'(x) is increasing

• f(x) is concave down & f'(x) is decreasing

If f(x) is INCREASING and CONCAVE DOWN

• f'(x) is positive and increasing

• f'(x) is positive and decreasing

• f'(x) is negative and increasing

• f'(x) is negative and decreasing

Tags

CCSS.HSF.LE.B.5

If f(x) has a max...

• f'(x) + -> -, f"(x) is +

• f'(x) + -> -, f"(x) is -

• f'(x) - -> +, f"(x) is +

• f'(x) - -> +, f"(x) is -

If f(x) changes from CONCAVE UP to CONCAVE DOWN

• f'(x) goes from + -> -

• f'(x) goes from - -> +

• f''(x) goes from + -> -

• f''(x) goes from - -> +

When f"(x) changes sign...

• f(x) has a POI

• f(x) has a max or a min

• f(x) has a critical value

• f'(x) has a POI

A critical value is when...

• f(x) is zero or undefined

• f'(x) is zero or undefined

• f(x) has a POI

• f"(x) changes sign

If f'(x) = 0 & f"(x) is NEGATIVE

• f(x) is concave up and has a max

• f(x) is concave up and has a min

• f(x) is concave down and has a max

• f(x) is concave down and has a min