For what values of x is the graph of the function f(x)= x^2e^x concave down. Give your answer using exact values
It will be concave down if f''(x) < 0
take the second derivative, and solve for f''(x) < 0
To determine when the graph of the function f(x) = x^2e^x is concave down, we need to find the interval where the second derivative of the function, f''(x), is negative.
Let's start by finding the first derivative, f'(x), of the function f(x) = x^2e^x. To do this, we can use the product rule:
f'(x) = (2x)(e^x) + (x^2)(e^x) = 2xe^x + x^2e^x
Next, we find the second derivative, f''(x), by differentiating f'(x) with respect to x:
f''(x) = (2x)(e^x) + (2)(e^x) + (2x)(e^x) + (x^2)(e^x) = 4xe^x + 2e^x + x^2e^x
Now we need to solve the inequality f''(x) < 0. Rearranging the terms gives us:
4xe^x + 2e^x + x^2e^x < 0
Factoring out e^x from the terms involving x yields:
(4x + 2 + x^2)e^x < 0
Since e^x is always positive, we can ignore it when determining the sign of the inequality. Thus, we solve the inequality:
4x + 2 + x^2 < 0
Rearranging:
x^2 + 4x + 2 < 0
Unfortunately, this quadratic inequality does not have an exact solution. However, we can determine the approximate solution by applying the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 4, and c = 2. Plugging these values into the quadratic formula, we get:
x = (-4 ± √(4^2 - 4(1)(2))) / (2(1))
x = (-4 ± √(16 - 8)) / 2
x = (-4 ± √8) / 2
x = (-4 ± 2√2) / 2
Simplifying further:
x = -2 ± √2
Therefore, the graph of the function f(x) = x^2e^x is concave down for x values in the interval (-2 - √2, -2 + √2) using exact values.