Find the maximum and minimum points on the graph of the function $f(x)=x^3-6x^2+9x+15$.

Step 1 ：The maximum and minimum points of a function occur at the points where the derivative of the function is equal to zero. So, first we find the derivative of the function $f(x)=x^3-6x^2+9x+15$.

Step 2 ：The derivative of $f(x)=x^3-6x^2+9x+15$ is $f^{\prime }(x)=3x^2-12x+9$.

Step 3 ：Setting $f^{\prime }(x)=0$, we get $3x^2-12x+9=0$. Solving this quadratic equation, we get two roots: $x=1$ and $x=3$.

Step 4 ：To determine whether these points are maximum or minimum, we check the second derivative of the function at these points. The second derivative of the function is $f^{″}(x)=6x-12$.

Step 5 ：Substituting $x=1$ into $f^{″}(x)$, we get $f^{″}(1)=-6<0$. This means that $x=1$ is a maximum point.

Step 6 ：Substituting $x=3$ into $f^{″}(x)$, we get $f^{″}(3)=6>0$. This means that $x=3$ is a minimum point.

Step 7 ：Finally, substitute $x=1$ and $x=3$ into the original function to find the y-coordinates of the maximum and minimum points: $f(1)=1^3-6\ast 1^2+9\ast 1+15=19$ and $f(3)=3^3-6\ast 3^2+9\ast 3+15=9$.