consider the following function. g(x)= -x² + 11x + 4 find the first and second derivatives of the function…

consider the following function. g(x)= -x² + 11x + 4 find the first and second derivatives of the function. g(x)= g(x)= determine where the function is concave upward and where it is concave downward. (enter your answer using interval notation. if an answer does not exist, enter dne.) concave upward concave downward

consider the following function. g(x)= -x² + 11x + 4 find the first and second derivatives of the function. g(x)= g(x)= determine where the function is concave upward and where it is concave downward. (enter your answer using interval notation. if an answer does not exist, enter dne.) concave upward concave downward

Answer

Explanation:

Step1: Find the first - derivative

Use the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $g(x)=-x^{2}+11x + 4$, we have $g'(x)=\frac{d}{dx}(-x^{2})+\frac{d}{dx}(11x)+\frac{d}{dx}(4)$. So $g'(x)=-2x + 11$.

Step2: Find the second - derivative

Differentiate $g'(x)=-2x + 11$ with respect to $x$. Using the power rule again, $g''(x)=\frac{d}{dx}(-2x)+\frac{d}{dx}(11)=-2$.

Step3: Determine concavity

Since $g''(x)=-2<0$ for all real $x$, the function is concave downward on $(-\infty,\infty)$ and concave upward on DNE.

Answer:

$g'(x)=-2x + 11$ $g''(x)=-2$ concave upward: DNE concave downward: $(-\infty,\infty)$