Q,18 - Calculus

Question

For Exercises 15–20, sketch the graph of a function f that has the indicated characteristics. If a graph is not possible, explain why. 

f positive, f' negative, and f'' positive on$\mathrm{ℝ}$.

Step-by-Step Solution

Answer

The possible graph of f is the following. 

Step 1. Given information.

The given interval is $\mathrm{ℝ}.$

f is positive. on interval

f' is negative on interval

f'' is positive on interval.

Step 2. sign chart of function.

Suppose both f and f' are differentiable on an interval I.

(a) If f'' is positive on I, then f is concave up on I.

(b) If f'' is negative on I, then f is concave down on I.

So sign chart of functions follows.

sign chart state that the f is increasing on interval.

Information of f'' did not properly state the graph of f.

antiderivative of f' is in the form of $f\left(x\right)+c.$

so we will do a vertical shift of f.

Step 3. Possible graph of functions.

The possible graph of f'' follows.

The possible graph of f follows.