Question

 

(a) How do you find the second derivative of a function?

(b) If both  are positive numbers, what does that tell you about the function itself?

(c) If we know that a function is increasing, but at a decreasing rate, what does that say about the first and second derivatives?

(d) Come up with a function where the first and second derivatives are non-zero but the third derivative is zero.

Answer 1

a) In order to find second derivative of function we first need to find first derivative of a function and then again find derivative of first derivative.

For example if 'f' is the function, then first derivative of 'f' is defines as:

Now derivative of above obtained derivative:

Second derivative of f:

b) If first derivative is positive, it indicates that function is increasing. If second derivative is positive then it indicates that function is concave up. Hence if both first derivate and second derivative are positive then it indicates that function is increasing and concave up.

c) If function is increasing then we know that its first derivative is positive. Here function is increasing but at a decreasing rate. (Note that derivative of any function shows its rate of increasing or decreasing) Therefore if we find derivate of first derivative it will show negative sign since rate is decreasing. That means function is concave down.

Therefore, in this case first derivate is positive, second derivative is neagtive. Function is increasing but its shape is concave down.

d) Derivative of any constant function is zero. Therefore in order to get third derivative zero we have to set a function such as its second derivative will be constant function. If we consider a polynomial function with degree 2. Its first derivative will have degree one. and second derivative will be constant term. Therefore, third derivative will be zero.

For example:

First derivative of f(x):

Second derivative of f(x):

Thirs derivative of f(x):