Question

Which of the following is not correct regarding signs of derivatives?

(a) If \(\frac{dy}{dx}>0,\) then the curve is increasing in [a, b]

(b) If \(\frac{dy}{dx}<0,\) then the curve is increasing in [a, b]

(c) If \(\frac{d^2 x}{dy^2}>0, \) then curve is convex and concave upward (holds water)

(d) If \(\frac{d^2 x}{dy^2}>0, \) then curve is concave downward (spills water)

This question was addressed to me in homework.

The doubt is from Curves in Cartesian Form topic in section Curve Tracing of Engineering Mathematics

Answer 1

Right choice is (b) If \(\frac{dy}{dx}<0,\) then the curve is increasing in [a, b]

For explanation: In an interval, a≤x≤b,

If \(\frac{dy}{dx}>0, \) then the curve is increasing in [a, b]

 If \(\frac{dy}{dx}< 0, \) then the curve is increasing in [a, b]

 If \(\frac{d^2x}{dy^2}>0, \) then curve is convex and concave upward (holds water)

 If \(\frac{d^2x}{dy^2}>0, \) then curve is concave downward (spills water)