Give an example of a function that has a local maximum at 33

Give an example of a function that has a local maximum
at (3,-3) and one which has a local minimum at (3.-3).

Solution

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A function f(x) has an extreme point when f\'(x) = 0. To find the x-coordinate, the equation f\'(x) = 0 has to be solved. If f\'\'(x) for the value of x obtained is positive, the point has a minimum value, else if it is negative the point has a maximum value.

f\'\'(3) has to be positive. Let it be 2.

f\'(x) = 0 should give a solution of x = 3, an example of this is f\'(x) = 2x - 6.

A function which has a value of -3 at x = 3 is x^2 - 6x + 6

The function f(x) = x^2 - 6x + 6 has a local minimum at (3, -3).

f\'\'(3) has to be negative. Let it be -2.

f\'(x) = 0 should give a solution of x = 3, an example of this is f\'(x) = -2x + 6.

A function which has a value of -3 at x = 3 is -x^2 + 6x -12

The function f(x) = -x^2 + 6x -12 has a local maximum at (3, -3).

A function with a local maximum at (3, -3) is f(x) = -x^2 + 6x -12 and a function with a local minimum at (3, -3) is f(x) = x^2 - 6x + 6