Calculus Homework

Question

A) Consider the following. f(x)=e^x + 3x

Find an interval, of length 1 and having integer endpoints, on which the function has a root. [-2,3] [-1,0]

[0,1]

[-1,1]

[2,3]

B) Consider the following. f(x)= x^3+5x-21

Find an interval, of length 1 and having integer endpoints, on which the function has a root. [-4,-3]

[-3,-2]

[1,2]

[2,3]

[3,4]

Answer 1

The idea is to find an interval where the function value at the left-hand endpoint has the opposite sign of the function value at the right-hand endpoint (and of course the interval is to be of length one.

Assuming the post is showing a selection of multiple choices:

For part A the following intervals can be immediately eliminated because they are not length one:

[-2, 3], [-1,1].

Let's check [-1, 0].

f(-1) = e^-1 + 3(-1) = 1/e -3 , Since e is about 2.7 1/e is positive and certainly less then 3 in absolute value, so f(-1) is negative.

f(0) = e⁰ + 3(0) = 1 which is positive.

Since f(x) is continuous the function must take on all values between f(-1) and f(0) which includes 0. So f has a root between -1 and 0.

Here is the idea for the other two remaining intervals:

[0,1]

f(0) = 1

f(1) = e + 3 which is also positive.

So the function does not have to have a value of 0 for x values between 0 and 1.

The pattern is similar for the interval [2,3], i.e. the function values at the endpoints are both positive.

Here is a Desmos graph confirming the above.

desmos.com/calculator/rwgbokgnwh

And based on the above, I am sure you can work out part B.