True or false... please explain why if false Let the function f be differentiable on an interval I containing c. If f has a maximum value at x = c, then f has a minimum value at x = c. A cubic function defined by f(x)= ax^3+bx^2+cx+d where a does not equal 0 will always have exactly three critical numbers

False.

Explanation:

1. Regarding the Maximum and Minimum Values:

  - The statement that if a function has a maximum value at x = c, then it also has a minimum value at x = c is generally not true. A function can have a maximum value at a specific point without necessarily having a minimum value at the same print. For example, consider the function defined by f(x) = -x^2, which has a maximum value at x = 0 but no minimum value.

2. Regarding Cubic Functions and Critical Numbers:

  - A cubic function defined by \(f(x) = ax^3 + bx^2 + cx + d\) will not always have exactly three critical numbers. Critical numbers occur where the derivative of the function is equal to zero or is undefined. For a cubic function, the derivative is a quadratic function, and it can have one, two, or three real roots (depending on the coefficients a, b, and c). Each real root corresponds to a critical number.

In summary, the first statement is generally false, and the second statement is false because the number of critical numbers for a cubic function can vary based on the coefficients in the function.