Question

Let f(x)= eˣ / x³. (c) Find, correct to two decimal places, the local minimum value of f and the value of x at which it occurs.

Answer 1

To find the local minimum value of f(x) = eˣ / x³ and the x-value at which it occurs, we need to find the critical points of the function and check if the endpoints provide any additional critical points.

Step-by-step explanation:

To find the local minimum value of f(x)= eˣ / x³ and the value of x at which it occurs, we need to find the critical points of the function. A critical point is where the derivative of the function is equal to zero or undefined.

To find the derivative, we can use the quotient rule. Let's denote u as eˣ and v as x³. The derivative of f(x) is:

f'(x) = (u'v - uv') / v²

Next, we set the derivative equal to zero and solve for x to find the critical points. Since the function is not defined for x = 0, we exclude that value.

We also check the endpoints of any closed interval to see if they provide any additional critical points. In this case, we only have one endpoint, which is infinity (as x approaches infinity or negative infinity).

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