Derivatives & the Shapes of Graphs

Question

1. Answer the questions below based on the following information about the function f . You must justify your
answers.
(i) The function f is continuous and differentiable for all values of x.
(ii) f(x) < 0 for x < 0; f(x) > 0 for 0 < x.
(iii) f'(x) < 0 for −6 < x < −2 and 5 < x.
(iv) f'(x) > 0 for x < −6 and −2 < x < 5.
(v) f''(x) < 0 for x < −4 and 3 < x < 7.
(vi) f''(x) > 0 for −4 < x < 3 and 7 < x

 

(a) On which intervals is the function decreasing?
(b) What is the x-coordinate of each local maximum (if any)?
(c) On which intervals is the function concave up?
(d) What is the x-coordinate of each inflection point (if any)?

 

Would someone be able to explain this to me?

Answer 1

A function is decreasing where its first derivative is negative; increasing where its first derivative is positive.

 

A local max occurs when the first derivative changes from positive to negative; a local min when it changes from negative to positive.

 

A function is concave up where its second derivative is positive; concave down where its second derivative is negative.

 

An inflection point occurs where the function changes concavity.

 

Hopefully this info will allow you to solve the problem.