Question

Use the first derivative test to find the critical points of the function f(x)=x⁷e⁴ˣ+3. What is the correct statement regarding the critical points?

A) There are no critical points for this function.
B) The function has one critical point at x=0.
C) The function has one critical point at x=-1/4.
D) The function has two critical points at x=0 and x=-1/4.

Answer 1

To find the critical points of the function f(x) = x⁷e⁴ˣ+3 using the first derivative test, we need to find the derivative of the function and set it equal to zero. The correct statement regarding the critical points is that there are no critical points for this function.

Step-by-step explanation:

To find the critical points of the function f(x) = x⁷e⁴ˣ+3 using the first derivative test, we need to find the derivative of the function and set it equal to zero. The first derivative of f(x) is f'(x) = 7x⁶e⁴ˣ+3 + 4x⁷e⁴ˣ+3 ln(e). To find the critical points, we set f'(x) = 0 and solve for x.

7x⁶e⁴ˣ+3 + 4x⁷e⁴ˣ+3 ln(e) = 0

Since there is an exponential term involved, we cannot solve this equation analytically. We can use numerical methods to approximate the critical point(s) of the function.

Therefore, the correct statement regarding the critical points is A) There are no critical points for this function.