f(x)=xe^(10x) Find the largest interval on which f(x) is concave upward.

Question

f(x)=xe^(10x) Find the largest interval on which f(x) is concave upward.

Answer 1

Hi,

In general, to find where a function is concave upward, we must find where its second derivative is positive (this is usually taken as the definition for a function to be concave up). So for this problem, we will begin by computing the second derivative.

f(x) = x·e^10x, so to find f'(x) we must apply the product rule and the chain rule. Using these rules, we obtain f'(x) = e^10x + 10x·e^10x. To find f''(x), we now take the derivative of f'(x), again applying the product rule and the chain rule. We then get that f''(x) = 10e^10x + 10e^10x + 100x·e^10x = 20e^10x(1 + 5x).

We wish to determine when this function is positive. We can begin by noting that 20e^10x is always positive, so the product 20e^10x(1 + 5x) is positive if and only if 1 + 5x is also positive. For 1 + 5x to be positive, we see that x must be larger than -1⁄5, because 1 + 5x > 0 implies 5x > -1, so x > -1⁄5. So we know our given function is concave up for all values of x greater than -1⁄5, and we conclude that the largest interval on which this function is concave up is (-1⁄5, ∞).

Thanks,

Aaron