find the following for the functions

Question

Answer the following questions of the function

f(x)=xsqrtx^2+25

on interval [-6,4]

a. f(x) is concave down

b. f(x) is concave up

c. the minimum of the function occurs at

d. The maximum of this function occurs at

Answer 1

Hi June!

So to start this problem, we are going to take the second derivative of the given equation:

f(x)= x√(x²+25)

Use the product and chain rule to get:

f'(x)= x²/√(x²+25) + √(x²+25)

Multiple equation by √(x²+25)/√(x²+25) to simplify:

f'(x)= 2x²+25/√(x²+25)

That's just the first derivative. Now get the second derivative:

f''(x)= x(2x²+75)/(x²+25)^3/2

Now we set f''(x)=0 to find the inflection points (the x value on the graph where the equation switches concavity).

x(2x²+75)/(x²+25)^3/2=0

Solving for x, we get x=0.

Now we notice that the interval is [-6,4], and our inflection point 0 falls in this interval.

a and b :

Now plug in a number between -6 and 0 to see the concavity of that section of the graph. A negative number will indicate concave down and a positive number will indicate the graph is concave up.

Any number between -6 and 0 will work.

Let's pick an easy number: -1. When x=-1, x(2x²+75)/(x²+25)^3/2= -0.58.

-0.58 is a negative number, so the function is concave down on the interval [-6,0] (we use a bracket next to 0 because the function is continuous at 0)

Now let's try the other side of the inflection point. What is the concavity between 0 and 4? Let's use another easy number on that interval, 1. When x=1, x(2x²+75)/(x²+25)^3/2= 0.58. 0.58 is a positive number, so the function is concave up on the interval [0,4].

c and d: I assume the question is asking for a local maximum and local minimum, since the absolute maximum and minimum would be the end points of the interval given.

Take the first derivative of the equation that we found in the beginning:

f'(x)= (2x²+25)/√(x²+25) and set it equal to 0 and solve for x.

You will get 2 values. You can tell which is the local maxima and minima by seeing which interval it falls on (the ones you determined were concave up or concave down in a and b)