the questions in level 1 are introductory problems. the hints contain links to videos covering this content…
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
Answer
Brief Explanations:
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)