Question

Find the maximum and minimum of the following functions: 

y = 5x³ + 2x² – 3x

Answer 1

y = 5x³ + 2x² – 3x

∴ the roots of dy/dx = 0 are x₁ = - 3/5 and x₂ = 1/3.

Method 1 (Second Derivative Test) :

∴ by the second derivative test, y is maximum at x = - 3/5 and maximum value of y at x = - 3/5

Hence, the function has maximum value 36/25 at x = - 3/5 and minimum value - 16/27 at x = 1/3.

Method 2 (First Derivative Test) :

Let h be a small positive number. Then

as h is small positive number.

∴ by the first deritative test, y is maximum at x = - 3/5 and maximum value of y at x = - 3/5

Let h be a small positive number. Then

∴ by the first derivative test, y is minimum at x = 1/3 and minimum value of y at x = 1/3

Hence, the function has maximum value 36/25 at x = - 3/5 and minimum value - 16/27 at x = 1/3.