Question

Answer the following questions for the function f(x)=x √(x²+4) defined on the interval -7 ≤ x ≤ 5. f(x) is concave down on the interval x=__ to x=___ f(x)

Answer 1

The function f(x)=x√(x²+4) is concave down on the entire interval -7 ≤ x ≤ 5.

Step-by-step explanation:

The function f(x)=x√(x²+4) is defined on the interval -7 ≤ x ≤ 5. To find the interval on which f(x) is concave down, we need to examine the second derivative of f(x). If the second derivative is negative, then the function is concave down.

First, let's find the second derivative of f(x):

f''(x) = 2x/(x²+4)^(3/2)

Now, we can find the critical points of f''(x) by setting it equal to zero and solving for x:

2x/(x²+4)^(3/2) = 0

Since the denominator cannot be zero, the equation has no solutions. Therefore, there are no critical points and f(x) is concave down on the entire interval -7 ≤ x ≤ 5.