Final answer:
To find the local minimum of the function f(x) = x⁴ - 12x² - 10, we differentiate, set the result equal to zero to find the critical points, and then use the second derivative to determine concavity. The function has local minima at x = √6 and x = -√6, both with a minimum value of -22.
Step-by-step explanation:
To find the local minimum of the function f(x) = x⁴ − 12x² − 10, first we need to find the critical points of the function by taking its derivative and setting it equal to zero. The derivative of f(x) is f'(x) = 4x³ − 24x. Setting the derivative equal to zero gives us the equation 4x³ − 24x = 0. Factoring out 4x leads to 4x(x² − 6) = 0, which has solutions x = 0, x = √6, and x = −√6.
Next, we need to determine which of these critical points are minima by evaluating the second derivative f''(x) = 12x² − 24 at each critical point. The second derivative at x = 0 is negative, indicating a local maximum. However, at x = √6 and x = −√6, the second derivative is positive, indicating a concave up shape and therefore local minima at these points.
Finally, we evaluate f(x) at x = √6 and x = −√6 to find the actual minimum values. The evaluations yield f(√6) = (√6)⁴ − 12(√6)² − 10 = −22 and f(−√6) = (−√6)⁴ − 12(−√6)² − 10 = −22. Thus, the local minimum value of the function is −22, and it occurs at both x = √6 and x = −√6.