#### 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

**Note:**

The derivative is zero where the function has a horizontal tangent.

**Graph:**

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

**Observation:**

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

By note, c2 is the derivative of the graph curve c1.

Thus, c1=f and c2=f′.

**Calculation:**

From Figure 1, f′(−1)>0 as the graph of f′ lies above the *x-*axis for x=−1.

Thus, f′(−1)>0 (1)

From Figure 1, f′(x) has horizontal tangent at x=1.

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

Thus, f″(1)=0 (2)

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

Compare f′(−1)>0 and f″(1)=0 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).