Second+Derivative

Are we getting faster or slower?

The Second Derivative is the derivative of the slope of a function. It measures how the rate of change is changing itself.

Formula: 

Ex: f(x)= 3x^3+2x^2 f'(x)= 9x^2+4x f''(x)= 18x+4

When looking at a graph, the second derivative relates with concavity. A function with a negative slope will curve downwards, while a positive function curves upwards.

"Concave up is part of a cup. Concave down is part of a frown"

A point of inflection is where the graph changes from concave up to concave down, or vise versa.

 The first derivative gives us the minimum and maximum points of the original function. The second derivative gives us the minimum and maximum points of the first derivative.

y = x^ 3 - 3x - 2  is  the function in blue <span style="font-family: Georgia,serif;">, <span style="color: #f31b1b; font-family: Georgia,serif;">y' = 3x ^2 - 3 <span style="font-family: Georgia,serif;">in red, and <span style="color: #1e730d; font-family: Georgia,serif;">y''=6x <span style="font-family: Georgia,serif;"> in green. The relationship between the three graphs shows in where they cross the x-axis and when they become concave up or down.

<span style="font-family: Georgia,serif;">Problem: <span style="font-family: Georgia,serif;">f(x) = 2x^3-3x^2-4. Use the second derivative test to classify the critical points.

<span style="font-family: Georgia,serif;">Step one: find first derivative of f(x). <span style="font-family: Georgia,serif;">Step two: find where f'(x)=0. <span style="font-family: Georgia,serif;">Step three: find second derivative of f(x). <span style="font-family: Georgia,serif;">Step four: plug in zeros found in step two into second derivative and solve to find critical points.

<span style="font-family: Georgia,serif;">If f''(x) > 0, f(x)is a local minimum. <span style="font-family: Georgia,serif;">If f''(x) < 0, f(x) is a local maximum. <span style="font-family: Georgia,serif;">If f''(x) = 0, then the test is inconclusive. f(x) could be a local maximum, local minimum, or neither.

<span style="font-family: Georgia,serif;">1) f'(x)= 6x^2-6x, <span style="font-family: Georgia,serif;">2) f'(x)=0 at x=0 and x=1, because 6(0)^2-6(0)=0 and 6(1)^2-6(1)=0 <span style="font-family: Georgia,serif;">3) f(x)=12x-6 <span style="font-family: Georgia,serif;">4) f(0)=-6 and f''(1)=6 <span style="font-family: Georgia,serif;">So, there's a local max at x=0 and a local min at x=1