Question

Let f(x) = ∫(from x to 2) -3x - 2eᵗ²) dt. At what value of x is f(x) a minimum?

a) -1
b) 0
c) 1
d) 2

Answer 1

To find the value of x at which f(x) is a minimum, we need to find the critical points of f(x).

Step-by-step explanation:

To find at what value of x the function f(x) = ∫(from x to 2) -3x - 2eᵗ²) dt is a minimum, we need to find the critical points of f(x) and determine whether they are minimum points.

First, we find the derivative of f(x) with respect to x, using the Fundamental Theorem of Calculus. The derivative of the integral with respect to x is the function inside the integral, evaluated at the upper limit of integration:

f'(x) = -3x - 2eᵗ² | x=2 = -3x - 2e⁴

Setting f'(x) = 0 and solving for x, we have:

-3x - 2e⁴ = 0

Solving this equation will give us the value of x at which f(x) has a minimum.