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 negative, f' positive, and f'' positive on $[0,3].$

Short answer

The possible graph of f is the following.

Step-by-step solution

Step 1. Given information. 

The given interval is $[0,3].$

f is negative on interval.

f'is positive on the 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 decreasing 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 off''follows.

The possible graph of f follows.