reletive extrema are where the first derivitive is 0 and the sign of the first derivitive changes
inflection points are where the 2nd derivitive is 0 and the sign of the 2nd derivitive changesd
I use f'(x)=1st derivitive and f''(x)=2nd derivitive
remember power rule: [tex]\frac{d}{dx} x^m=mx^{m-1}[/tex]
[tex]f(x)=9x^\frac{1}{3}+\frac{9}{2}x^\frac{4}{3}[/tex]
f'(x)
[tex]f'(x)=(9)(1/3)x^\frac{-2}{3}+(\frac{9}{2})(\frac{4}{3})x^\frac{1}{3}[/tex]
[tex]f'(x)=3x^\frac{-2}{3}+6x^\frac{1}{3}[/tex]
f''(x)
[tex]f''(x)=(3)(\frac{-2}{3})x^\frac{-5}{3}+(6)(\frac{1}{3})x^\frac{-2}{3}[/tex]
[tex]f''(x)=-2x^\frac{-5}{3}+2x^\frac{-2}{3}[/tex]
I'm not going to show the work for solving for when f'(x)=0 and f''(x)=0 because at this stage of the game, you should be able to do that easily
f'(x) is equal to 0 when x=-0.5, the sign changes from negative to positive at this point so at this point, the function has a relative minimum
f''(x) is equal to 0 when x=1, the sign changes from negative to positive at this point so at this point, the function is changing from concave down to concave up
Answer:
f(x) = 9x∧1/3 + 91/2x∧4/3
Step-by-step explanation: y=f(x)
∴ y = 9x∧1/3 + 91/2x4/3
dy/dx = {9x∧-2/3}/3 + {36x∧1/3}/3
dy/dx = 3/x∧2/3 + 12x∧1/3 = 3/∧x2/3{ 1 + 4x}
For, 1 +4x = 0
4x = -1
∴ x = -1/4
Inflection point, x = -1/4
