Find the local maxima and minima of a function.

Question

The question states: "You are given the function f(x) = e^x(x+3). Find the local minima and maxima."

The concept baffles me, and any explanation alongside the solution would be great.

My rule knowledge includes:

If the function f is differentiable at x0 and has a local minima or maxima at x0 then f'(x0) = 0

Answer 1

If f(x) is the function, then the derivative, f '(x) is the slope of the function at any value of "x". If you consider slope, a zero slope is a horizontal line that would occur when a function graph changes directions (or has a min or max value). So taking the derivative and setting it equal to zero finds the values of "x" that have zero slope (minimums or maximums). Of course, there is another possibility. The curve COULD be transitioning from a concave up to a concave down condition and also have a zero slope at that transition point (also called a point of inflection). So, we must check in some fashion to ensure the point is actually a min or max.

So for f(x) = (e^x)(x + 3), first let's consider if the domain is good for all real numbers. e^x is ok and so is (x + 4) so there are no places where the function is undefined.

To find f '(x), we must use the product rule (u•v)' = u'v + uv'

Let u = e^x then u' = e^x and let v = x + 3 so v' = 1

So f '(x) = e^x(x + 3) + e^x(1) = e^x[(x + 3) + 1] = e^x(x + 4)

Again, there are no values of "x" where the derivative is undefined.

Setting f' equal to zero:

e^x(x + 4) = 0

Set each piece equal to zero:

e^x = 0 which never occurs and

(x + 4) = 0 which occurs at x = -4

So the only POSSIBLE min or max occurs at x = -4

To determine if we have a min, a max, or a point of inflection, we can consider the slope on each side of x = -4

At x = -2, the slope is f '(-2) = e^-2(-2 + 4) = 2/e² ≈ 0.27 (a positive number)

At x = -6, the slope is f '(-6) = e^-6(-6 + 4) = -2/e⁶ ≈ -0.005 (a negative number)

So, since there is a negative slope on the left of x = -4 and a positive slope on the right of x = -4, then x = -4 must be a minimum. There are no maximums.