Answer:
We have an extrema (local minimum) at x = -0.125
An inflection point at x = 0.25
Step-by-step explanation:
The given function is given as follows;
![f(x) = 3x^{1/3} + 6x^{4/3}](https://tex.z-dn.net/?f=f%28x%29%20%3D%203x%5E%7B1%2F3%7D%20%2B%206x%5E%7B4%2F3%7D)
At the extrema points, f'(x) = 0 which gives;
![0 = \dfrac{\mathrm{d} \left (3x^{1/3} + 6x^{4/3} \right )}{\mathrm{d} x} = \dfrac{(8 \cdot x+1) \times \sqrt[0.3]{x} }{x}](https://tex.z-dn.net/?f=0%20%3D%20%5Cdfrac%7B%5Cmathrm%7Bd%7D%20%20%5Cleft%20%283x%5E%7B1%2F3%7D%20%2B%206x%5E%7B4%2F3%7D%20%20%5Cright%20%29%7D%7B%5Cmathrm%7Bd%7D%20x%7D%20%3D%20%5Cdfrac%7B%288%20%5Ccdot%20x%2B1%29%20%5Ctimes%20%5Csqrt%5B0.3%5D%7Bx%7D%20%20%7D%7Bx%7D)
(8x + 1) =x- (0/((x)^(1/0.3)) = 0
x = -1/8 = -0.125
f''(x) gives;
![f''(x) = \dfrac{\mathrm{d} \left (\dfrac{(8 \cdot x+1) \times \sqrt[0.3]{x} }{x} \right )}{\mathrm{d} x} = \dfrac{ \left (\dfrac{8}{3}\cdot x^2 - \dfrac{2}{3} \cdot x \right ) \times \sqrt[0.3]{x} }{x^3}](https://tex.z-dn.net/?f=f%27%27%28x%29%20%3D%20%5Cdfrac%7B%5Cmathrm%7Bd%7D%20%20%5Cleft%20%28%5Cdfrac%7B%288%20%5Ccdot%20x%2B1%29%20%5Ctimes%20%5Csqrt%5B0.3%5D%7Bx%7D%20%20%7D%7Bx%7D%20%5Cright%20%29%7D%7B%5Cmathrm%7Bd%7D%20x%7D%20%3D%20%5Cdfrac%7B%20%5Cleft%20%28%5Cdfrac%7B8%7D%7B3%7D%5Ccdot%20x%5E2%20-%20%5Cdfrac%7B2%7D%7B3%7D%20%5Ccdot%20x%20%5Cright%20%29%20%5Ctimes%20%5Csqrt%5B0.3%5D%7Bx%7D%20%7D%7Bx%5E3%7D)
Substituting x = -0.125 gives f''(x) = 32 which is a minimum point
The inflection point is given as follows;
![\dfrac{ \left (\dfrac{8}{3}\cdot x^2 - \dfrac{2}{3} \cdot x \right ) \times \sqrt[0.3]{x} }{x^3} = 0](https://tex.z-dn.net/?f=%5Cdfrac%7B%20%5Cleft%20%28%5Cdfrac%7B8%7D%7B3%7D%5Ccdot%20x%5E2%20-%20%5Cdfrac%7B2%7D%7B3%7D%20%5Ccdot%20x%20%5Cright%20%29%20%5Ctimes%20%5Csqrt%5B0.3%5D%7Bx%7D%20%7D%7Bx%5E3%7D%20%3D%200)
![\dfrac{8}{3}\cdot x^2 - \dfrac{2}{3} \cdot x \right }{} = 0 \times \dfrac{x^3}{ \sqrt[0.3]{x}}](https://tex.z-dn.net/?f=%5Cdfrac%7B8%7D%7B3%7D%5Ccdot%20x%5E2%20-%20%5Cdfrac%7B2%7D%7B3%7D%20%5Ccdot%20x%20%5Cright%20%20%7D%7B%7D%20%3D%200%20%5Ctimes%20%5Cdfrac%7Bx%5E3%7D%7B%20%5Csqrt%5B0.3%5D%7Bx%7D%7D)
![\dfrac{8}{3}\cdot x - \dfrac{2}{3} \right }{} = 0](https://tex.z-dn.net/?f=%5Cdfrac%7B8%7D%7B3%7D%5Ccdot%20x%20-%20%5Cdfrac%7B2%7D%7B3%7D%20%20%5Cright%20%20%7D%7B%7D%20%3D%200)
x = 2/3×3/8 = 1/4 = 0.25
We check the value of f''(x) at x = 0.24 and 0.26 to determine if x = 0.25 is an inflection point as follows;
At x = 0.24, f''(x) = -0.288
At x = 0.26, f''(x) = 0.252
0.25 is an inflection point