Determine the nature and location of the stationary points of the function y = 5x^3 + 7x^2.?

2nd stationary point x = -14/15 , y = 5(-14/15)^3+7(-14/15)^2 = ...calculator

I had to study "table certain factors." not a undemanding term in my math practise interior the U.S., even though it capacity a factor the place the tangent is 0. discover the spinoff: y' = cosx - sinx and set to 0 to discover the table certain factors so cosx=sinx or tanx = a million. Tan x=a million for x = pi/4 At x = pi/4 the two sinx and cosx = sqrt(2)/2 so this is in all probability a optimal, with y = sqrt(2) we can examine to work out if it somewhat is a optimal by using looking the 2nd spinoff: y''= -sinx - cosx At x=pi/4, the two sinx and cosx are constructive, so y'' is adverse, and it somewhat is concave downward at this factor, so (pi/4, sqrt(2)) is a optimal discover factors of inflection: set y'' = 0. sinx = -cosx or tanx = -a million x = 3pi/4 is an inflection factor chart: 0<x< pi/4 pi/4 pi/4 =<x< pi/2 pi/2 pi/2 <x< 3pi/4 3pi/4 3pi/4<x< pi y'' - - - - - 0 + y' + 0 - - - - - y + sqrt2 + + + 0 - So, y starts off off constructive, reaches a max at x=pi/4, decreases (y'<0) to 3pi/4, after that's adverse. this is concave downward (y''<0) till 3pi/4, that's an inflection factor, whilst it adjustments to concave upward Hmm...seems my heavily designed chart would not line up as quickly as I submit it. you will could desire to place each and each mark in each and each row of the "chart" interior the ideal type and make your guy or woman chart.

dy/dx=15x^2+14x
At stationary point dy/dx=0
15x^2+14x=0
x(15x+14)=0
x=0 or x=-14/15

To find, you just need to substitute those x values into the original equation y=5x^3+7x^2
when x=0, y=0, when x=-14/15, y=1372/675

Therefore stationary points: (0, 0) and (-14/15, 1372/675). Good luck