How do you find the local maximum and minimum values of #f(x)=2x^3 + 5x^2 - 4x - 3#?

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Answer 1 · 2017-06-10T14:39:18

#f(-2)=9# is local Maxima, #f(1/3)=-100/27# is local

Minima.

We know that, for local Extreme values, #f'(x)=0.#

Also, #f''(x)<0# for Maxima, and, #f''(x) >0# for Minima.

#f(x)=2x^3+5x^2-4x-3#

# rArr f'(x)=6x^2+10x-4, &, f''(x)=12x+10.#

# f'(x)=0 rArr 2(3x^2+5x-2)=0.#

# rArr 2(x+2)(3x-1)=0.#

# rArr x=-2, x=1/3.#

Now, #f''(-2)=-24+10=-14 < 0.#

#:. f# has a local maxima at #x=-2,# &, it is,

#f(-2)=-16+20+8-3=9.#

Also, #f''(1/3)=4+10=14 > 0.#

#:. f# has a local minima at #x=1/3,# which is,

#f(1/3)=2/27+5/9-4/3-3=-100/27.#

Thus, #f(-2)=9# is local Maxima, #f(1/3)=-100/27# is local

Minima.

Enjoy Maths.!