Question

the maximum value of the Function f (x)=2x^3-15x^2+36x-48on the setA:(x belong tox^2+20 <=9x) is

Answer 1

x² + 20≤ 9x

x² -9x +20 ≤ 0

x² -4x -5x +20 ≤ 0

x( x -4) -5(x -4) ≤ 0

(x -4)(x -5) ≤ 0

4 ≤ x ≤ 5

now ,

f(x ) = 2x³ - 15x² +36x -48

differentiate wrt x

f'(x) = 6x² -30x +36 = 6(x²-5x +6)

= 6(x -2)(x -3)

you can see in the graph
at 4≤ x ≤ 5 function is increasing so,

maximum value of f(x) at x = 5

f(5) = 2(5)³-15(5)² +36(5) -48

= 250 -375 + 180 -48

= 430 - 423 = 7