for f(x)=2x²ln x - 5x² + 3x - 1, determine the interval (a,b) where f is concave down. as your answer…

Explanation:

Step1: Find the first - derivative

Using the product rule and power rule, if (y = 2x^{2}\ln x-5x^{2}+3x - 1). The derivative of (y = 2x^{2}\ln x) is (y^\prime_1=2(2x\ln x + x^{2}\cdot\frac{1}{x})=4x\ln x+2x) by the product rule ((uv)^\prime = u^\prime v+uv^\prime) where (u = 2x^{2}) and (v=\ln x). The derivative of (-5x^{2}) is (-10x), the derivative of (3x) is (3) and the derivative of (- 1) is (0). So (f^\prime(x)=4x\ln x+2x - 10x + 3=4x\ln x-8x + 3).

Step2: Find the second - derivative

Using the product rule on (4x\ln x) (where (u = 4x) and (v=\ln x)), (u^\prime=4) and (v^\prime=\frac{1}{x})). (f^{\prime\prime}(x)=4(\ln x + x\cdot\frac{1}{x})-8=4\ln x - 4).

Step3: Find where (f^{\prime\prime}(x)<0)

Set (f^{\prime\prime}(x)<0), so (4\ln x-4 < 0). Add (4) to both sides: (4\ln x<4). Divide both sides by (4): (\ln x < 1). Since (y = \ln x) and (y = 1), and (y=\ln x) is an increasing function, (x<e). Also, the domain of (f(x)) is (x>0) (because of (\ln x)). So the interval where (f(x)) is concave - down is ((0,e)), then (a = 0) and (b = e\approx2.71828).

Step4: Calculate (a + b)

(a + b=0 + e\approx2.718).

Answer:

(2.718)