#### To determine

**To Identify:** The bigger quantity among f′(−1) and f′′(1) with proper explanation.

#### Answer

The value of f′′(1) is bigger than f′(−1).

#### Explanation

**Graph:**

The given graph of *f* and f′ is shown in Figure 1.

**Observation:**

From the Figure 1, it is observed that the point where c2 has functional value 0 is the same point at which the graph of c1 has horizontal tangent.

Thus, c1=f and c2=f′.

**Calculation:**

From the Figure 1, it is observed that f′(−1)<0 as the graph of f′ lies below the *x-*axis for x=−1.

Thus, f′(−1)<0. (1)

From the Figure 1, it is observed that f′(x) has a horizontal tangent at x=1.

Recall that the derivative of a function is zero where the function has a horizontal tangent.

Thus, the value of the derivative of f′ at x=1 is 0.

That is, f″(1)=0. (2)

From equation (1) and (2), f′(−1)<0 and f″(1)=0.

Compare f′(−1) and f′′(1) to identify which one is bigger

f′(−1)<0=f″(1)f′(−1)<f″(1)

Thus, the value of f′′(1) is bigger than f′(−1).