Answer :
Answer:
[tex]\begin{gathered} f\mleft(x\mright)\text{ has a relative minimum at x=1/4} \\ f\mleft(x\mright)\text{ has a relative maximum at x=0} \\ f\mleft(x\mright)\text{ has an inflection point at x=1/8} \end{gathered}[/tex]Step-by-step explanation:
To find the relative minimum and maximum, find the first derivate of the following function:
[tex]f\mleft(x\mright)=8x^3-3x^2[/tex]Derivating, using the derivative of a sum is equal to the sum of the derivatives. f'(x)=g'(x)+h'(x)
[tex]f^{\prime}\left(x\right)=24x^2-6x[/tex]A relative maximum point is a point where the function changes direction from increasing to decreasing. Similarly, a relative minimum point is a point where the function changes direction from decreasing to increasing.
Equalize the first derivable to 0.
[tex]\begin{gathered} 24x^2-6x=0 \\ 6x\left(4x-1\right)=0 \\ 6x_1=0 \\ x_1=0 \\ \\ 4x_2-1=0 \\ x_2=\frac{1}{4} \end{gathered}[/tex]Then, the x-coordinate of the relative minimum is x=1/4, and the x-coordinate of the relative maximum is x=0.
The inflection point is a point of a curve at which a change in the direction of curvature occurs. To find the inflection points, find the second derivate and solve for equals 0:
[tex]\begin{gathered} f^{^{\prime}^{\prime}}\left(x\right)=48x-6 \\ Equalize\text{ to 0:} \\ 48x-6=0 \\ x=\frac{6}{48} \\ x=\frac{1}{8} \end{gathered}[/tex]