A curve has equation: y = x^3 - 3x^2 + 5. Show that the curve has a minimum point when x = 2.

A minimum point will have a gradient of 0 (although so will a maximum point or a point of inflection). dy/dx = 3x2-6x. We can substitute x = 2 into this equation to give 0 (alternatively solve 3x2-6x = 0 to give x = 2 as a root). Thus the gradient at x = 2 is 0 so it is either a minimum point, maximum point or point of inflection. To prove it is in fact a minimum point, we can differentiate the original equation of the curve once more to give d2y/dx2= 6x - 6 (the second derivative). Substituting x = 2 into this equation gives a positive value. Positive value = minimum point; negative value = maximum point; zero value requires further investigation by finding gradients of x values slightly smaller and larger than 2 (this is also a possible alternative method to determine which type of stationary point it is to begin with).

TC
Answered by Tom C. Further Mathematics tutor

7934 Views

See similar Further Mathematics GCSE tutors

Related Further Mathematics GCSE answers

All answers ▸

express z(2+i)=(1+2i)^2 in the form z=x+iy


What is the equation of a circle with centre (3,4) and radius 4?


Find the coordinates of the minimum/maximum of the curve: Y = 8X - 2X^2 - 9, and determine whether it is a maximum or a minimum.


How can I find the equation of a straight line on a graph?


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences