Question

Sketch the graph of the function which satisfies the conditions \({f^\prime }(x) < 0\) and \({f^{\prime \prime }}(x) < 0\) for all \(x\).

Short answer

The sketch of the function that satisfies the given conditions is shown below in figure 1.

Step-by-step solution

Given data

The conditions \({f^\prime }(x) < 0\) and \({f^{\prime \prime }}(x) < 0\) for all \(x\).

Concept of concavity test and increasing /decreasing test

Concavity test:

(a) If\({f^{\prime \prime }}(x) > 0\)for all\(x\)in\(l\), then\(f\)is concave upward on\(l\).

(b) If\({f^{\prime \prime }}(x) < 0\)for all\(x\)in\(l\), then\(f\)is concave downward on\(l\).

Increasing/decreasing test:

(a) If\({f^\prime }(x) > 0\)on an interval, then\(f\)is increasing on that interval.

(b) If\({f^\prime }(x) < 0\)on an interval, then\(f\)is decreasing on that interval.

 Step 3: Sketch of the function that satisfies the given conditions

Since the first derivative of the function \(f(x)\) is negative, the function \(f(x)\) is always decreasing.

That is, if \({f^\prime }(x) < 0\), then \(f(x)\) is decreasing.

Since the second derivative of the function \(f(x)\) is negative, the function \(f(x)\) is always concave downward.

That is, if \({f^{\prime \prime }}(x) < 0\), then \(f(x)\) is concave downward.

Therefore, the sketch of the function that satisfies the given conditions is shown below in figure 1.

From figure 1, it is observed that the function is always decreasing and concave downward.