Question

the questions in level 1 are introductory problems. the hints contain links to videos covering this content. question 1 (1 point) if $f(x)>0$ for all x on an interval i, then f is increasing on i is decreasing on i has a local minimum on i has a local maximum on i view hint for question 1

Explanation:

According to the first - derivative test, if the derivative of a function \(f(x)\), \(f^{\prime}(x)>0\) for all \(x\) in an interval \(I\), then for any two points \(x_1\) and \(x_2\) in \(I\) with \(x_1 < x_2\), we have \(f(x_2)-f(x_1)=\int_{x_1}^{x_2}f^{\prime}(t)dt>0\) (by the fundamental theorem of calculus). This implies \(f(x_2)>f(x_1)\), so the function is increasing on the interval \(I\).
A function is decreasing when \(f^{\prime}(x)<0\) on an interval. For a local minimum, we need \(f^{\prime}(x)\) to change sign from negative to positive, and for a local maximum, \(f^{\prime}(x)\) to change sign from positive to negative. Since we only know \(f^{\prime}(x)>0\) (no sign change), the options about local minimum and maximum are incorrect.

Answer:

is increasing on \(I\)