Find the interval where k(x)=x^(2)e^(x) is concave down.

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Find the interval where  k(x)=x^(2)e^(x) is concave down.

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Find First Derivative: To determine where the function k(x) = x^(2)e^(x) is concave down, we need to find the second derivative of k(x) and then find the interval where this second derivative is negative.Find Second Derivative: First, let's find the first derivative k'(x) using the product rule, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Analyze Sign of Expression: Applying the product rule to k(x) = x^(2)e^(x), we get: k'(x) = d/dx(x^(2)) * e^(x) + x^(2) * d/dx(e^(x)) k'(x) = 2x * e^(x) + x^(2) * e^(x) k'(x) = e^(x)(2x + x^(2))Find Roots of Quadratic: Now, let's find the second derivative k''(x) by differentiating k'(x). We will again use the product rule.Apply Quadratic Formula: Differentiating k'(x) = e^(x)(2x + x^(2)), we get: k''(x) = d/dx(e^(x)) * (2x + x^(2)) + e^(x) * d/dx(2x + x^(2)) k''(x) = e^(x)(2x + x^(2)) + e^(x)(2 + 2x) k''(x) = e^(x)(2 + 4x + x^(2))Determine Concave Down Interval: To find where k''(x) is negative, we need to analyze the sign of the expression inside the parentheses, since e^(x) is always positive for all x.We need to find the roots of the quadratic equation 2 + 4x + x^(2) = 0 to determine the intervals where the expression is negative.The quadratic equation can be rewritten as x^(2) + 4x + 2 = 0. To solve for x, we can use the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a), where a = 1, b = 4, and c = 2.Plugging the values into the quadratic formula, we get: x = [-4 ± sqrt(4^2 - 4*1*2)] / (2*1) x = [-4 ± sqrt(16 - 8)] / 2 x = [-4 ± sqrt(8)] / 2 x = [-4 ± 2sqrt(2)] / 2 x = -2 ± sqrt(2)The roots of the quadratic equation are x = -2 + sqrt(2) and x = -2 - sqrt(2). The quadratic function is concave up, which means it is negative between its roots.Therefore, the function k(x) = x^(2)e^(x) is concave down on the interval (-2 - sqrt(2), -2 + sqrt(2)).

Find the interval where $k(x)=x^{2} e^{x}$ is concave down.

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Find the interval where $k(x)=x^{2} e^{x}$ is concave down.