How do you find the number and location of local minima of a given function?

[spaghetti3451]

Take the function

$$f(x) = ax^{2} + bx^{4} - c \cos(x/d),$$

where $a$, $b$, $c$ and $d$ are arbitrary parameters.

For some given choice of the parameters, how do you find the number of local minima of the function and the location of the minima?

 

[mmwave]

To find the number of local minima of the given function, we first need to take the derivative of the function and set it equal to zero. This will give us the critical points of the function, where the slope is equal to zero.

$$f'(x) = 2ax + 4bx^{3} + \frac{c}{d} \sin(x/d) = 0$$

We can then solve this equation for $x$ to find the critical points. Once we have the critical points, we can use the second derivative test to determine if they are local minima or maxima.

The second derivative of the function is given by:

$$f''(x) = 2a + 12bx^{2} + \frac{c}{d^{2}} \cos(x/d)$$

If $f''(x) > 0$, then the critical point is a local minimum. If $f''(x) < 0$, then the critical point is a local maximum. If $f''(x) = 0$, then the test is inconclusive and we need to try a different method to determine the nature of the critical point.

We can also use the first derivative test to determine if the critical points are local minima or maxima. This involves plugging in values on either side of the critical point into the first derivative and checking if the slope changes from positive to negative or vice versa. If the slope changes from positive to negative, then the critical point is a local maximum. If the slope changes from negative to positive, then the critical point is a local minimum.

To find the location of the local minima, we can plug in the values of $x$ that we found from solving the first derivative equation into the original function. This will give us the corresponding $y$ values of the local minima.

In summary, to find the number and location of the local minima of the given function, we need to:

1. Take the derivative of the function and set it equal to zero to find the critical points.
2. Use the second derivative test or the first derivative test to determine if the critical points are local minima or maxima.
3. Plug in the values of $x$ from the critical points into the original function to find the corresponding $y$ values of the local minima.