Question

a function is concave up and decreasing on an interval. what does this tell us about its slope?
a. the slope is positive and increasing
b. the slope is negative and decreasing
c. the function has a minimum at every point
d. the slope is negative but increasing

Explanation:

Step1: Recall function - decreasing property

A function $y = f(x)$ is decreasing on an interval if $f^{\prime}(x)<0$ on that interval. The first - derivative $f^{\prime}(x)$ represents the slope of the tangent line to the function. So, if the function is decreasing, the slope of the tangent line (the slope of the function) is negative.

Step2: Recall function - concave - up property

A function $y = f(x)$ is concave up on an interval if $f^{\prime\prime}(x)>0$ on that interval. The second - derivative $f^{\prime\prime}(x)$ represents the rate of change of the first - derivative $f^{\prime}(x)$. If $f^{\prime\prime}(x)>0$, then $f^{\prime}(x)$ (the slope of the function) is increasing.

Answer:

D. The slope is negative but increasing