Let f and g be two twice differentiable functions defined form R → R. If f (x) has a local maximum at x = c and g(x) has a local minimum at x = c, then*
Question
Let f and g be two twice differentiable functions defined from R → R.
If f(x) has a local maximum at x = c and g(x) has a local minimum at x = c, then *
Solution
The statement seems to be incomplete, but I can provide some information based on what you've given.
For a function to have a local maximum or minimum at x = c, the first derivative of the function at that point must be zero, i.e., f'(c) = 0 and g'(c) = 0. This is known as the first derivative test.
The second derivative test can then be used to determine whether these points are local maxima or minima. If the second derivative at that point is negative, f''(c) < 0, then the function has a local maximum at x = c. If the second derivative at that point is positive, g''(c) > 0, then the function has a local minimum at x = c.
So, if f(x) has a local maximum at x = c, then f'(c) = 0 and f''(c) < 0. If g(x) has a local minimum at x = c, then g'(c) = 0 and g''(c) > 0.
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