for f(x)=x^4 - 10x^3 + 24x^2 - 3, determine the interval (a,b) where f is concave down. as your answer…

for f(x)=x^4 - 10x^3 + 24x^2 - 3, determine the interval (a,b) where f is concave down. as your answer, please input a + b.

for f(x)=x^4 - 10x^3 + 24x^2 - 3, determine the interval (a,b) where f is concave down. as your answer, please input a + b.

Answer

Explanation:

Step1: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, we have $f(x)=x^{4}-10x^{3}+24x^{2}-3$. Then $f'(x)=4x^{3}-30x^{2}+48x$.

Step2: Find the second - derivative

Differentiate $f'(x)$ again. $f''(x)=12x^{2}-60x + 48$.

Step3: Set $f''(x)<0$ for concave - down

$12x^{2}-60x + 48<0$. Divide through by 12: $x^{2}-5x + 4<0$.

Step4: Factor the quadratic inequality

Factor $x^{2}-5x + 4$ as $(x - 1)(x - 4)<0$.

Step5: Solve the inequality

The roots of the equation $(x - 1)(x - 4)=0$ are $x = 1$ and $x = 4$. The solution of the inequality $(x - 1)(x - 4)<0$ is $1<x<4$. So $a = 1$ and $b = 4$.

Answer:

5