Answer to Question #147654 in Calculus for liam donohue

Question #147654

Answer the following True-False quiz. (Enter "T" or "F".)
1. A continuous function on a closed interval always attains a maximum and a minimum value.
2. Continuous functions are always differentiable.
3. If f′(c)=0 and f′′(c)>0, then f(x) has a local minimum at c.
4. If f(x)=e2, then f′(x)=2e.
5. If f(x) and g(x) are increasing on an interval I, then f(x)g(x) is increasing on I.
6. If a function has a local maximum at c, then f′(c) exists and is equal to 0.
7. Differentiable functions are always continuous.

Expert's answer

1. True because :

Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value both.

2. False because :

a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

3. True because :

The answer is that f ''(c) tells us the concavity of f(x) at x = c. If f(x) is concave up, i.e., if f ''(c) > 0, then the critical point should be a local minimum.

4. False because :

e²is a constant, so the derivative is zero.

5. False because :

f and g increasing on I means that f' > 0 on I and g' > 0 on I. However,

d/dx fg = f * g' + f' * g

and although we know g' and f' are guaranteed to be positive we don't have any guarantees about the sign of the values of the functions themselves. Basically, if at least one of the functions has negative values there is the potential that the derivative of the product will be negative.

6. True because :

Fermat's Theorem: If f has local maximum or local minimum at c, and f is differentiable at c, then f (c) = 0. This theorem says that if a function has a local maximum or local minimum at a point c, then its tangent at that point (if it has a tangent) must be a horizontal line.