Solution:

Step 1: Derive the function to find critical points

• Take the derivative of $f(x)=6x^4-32x^3+48x^2-7$.

• Apply the Power Rule: $\frac{d}{dx}{x}^n=n{x}^{n-1}$.

• Derivative: $f^{\prime }(x)=24x^3-96x^2+96x$.

Step 2: Find the second derivative for the concavity test

• Derive $f^{\prime }(x)$ to get $f^{″}(x)$.

• Second Derivative: $f^{″}(x)=72x^2-192x+96$.

Step 3: Solve $f^{\prime }(x)=0$ to find critical points

• Factor $f^{\prime }(x)$: $24x(x^2-4x+4)=24x(x-2{)}^2$.

• Set equal to zero: $x=0,2$.

Step 4: Determine the nature of critical points

• Use the second derivative test.

• For $x=0$: $f^{″}(0)=96>0$, so it's a local minimum.

• For $x=2$: $f^{″}(2)=0$, inconclusive, use the first derivative test.

Step 5: Apply the first derivative test for $x=2$

• Check sign changes of $f^{\prime }(x)$ around $x=2$.

• No sign change implies $x=2$ is neither a maximum nor a minimum.

Step 6: Find the y-values for the critical points

• For $x=0$: $f(0)=-7$.

• The local minimum is at $(0,-7)$.

Step 7: Conclusion

• The function has a local minimum at $(0,-7)$.

• No local maximum found.

Knowledge Notes:

1. Power Rule: Used to differentiate terms with powers of $x$. If $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$.

2. Product Rule: Used when differentiating products of functions. If $u(x)$ and $v(x)$ are functions of $x$, then the derivative of $u(x)v(x)$ is $u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$.

3. Sum Rule: Used when differentiating a sum of functions. The derivative of a sum is the sum of the derivatives.

4. Second Derivative Test: If $f^{″}(x)>0$, the function is concave up and the point is a local minimum. If $f^{″}(x)<0$, it's concave down and the point is a local maximum.

5. First Derivative Test: Used to determine if a critical point is a local maximum or minimum by analyzing the sign changes of the first derivative around the critical point.

6. Critical Points: Points where the first derivative is zero or undefined. They are potential locations for local maxima, minima, or points of inflection.

Solution:"The function $f(x)=6x^4-32x^3+48x^2-7$ has a local minimum at the point $(0,-7)$. No local maximum was found." Knowledge Notes:"The solution involves applying calculus concepts such as the Power Rule, Product Rule, Sum Rule, and the First and Second Derivative Tests to determine the local extrema of the function."