#### To determine

**To sketch:** The graph of the function f(x)=x4−3x3+6x2+7x+30.

#### Explanation

**Given:**

The function f(x)=x4−3x3+6x2+7x+30 in the screening rectangle [−3,5] by [−10,50].

**Graph:**

Use online graphing calculator and draw the graph of the given function as shown below in Figure 1.

From Figure 1, it is observed that the given polynomial function is differentiable everywhere.

#### To determine

**To estimate:** The slope of the function f(x) by using the graph obtained in part (a).

#### Explanation

The value of the derivative of the function at any point *x* can be estimated by drawing the tangent line at any point (x,f(x)) and then obtain the slope.

Mark the slope of the tangent as a point in *y*-axis and the value of *x* as a point in *x*-axis in the graph of f′(x).

Proceed in the similar way at several points and obtain the rough graph of the f′(x) as shown in Figure 2.

**Graph:**

The rough sketch of the derivative function, f′(x) in the graph of the function f(x) obtained in part (a) is shown below in Figure 2.

From Figure 2, it is observed that the slope of the tangent to the function is zero when the values of *x* approximately x1=−1.25 , x2=0.5 and x3=3.

The slope of tangent to the function is negative when x∈(−∞,x1)∪(x2,x3).

The slope of tangent to the function is positive when x∈(x1,x2)∪(x3,∞).

#### To determine

**To calculate:** The derivative of function f(x).

#### Answer

The derivative of f(x)=x4−3x3+6x2+7x+30 is 4x3−9x2+12x+7_.

#### Explanation

**Derivative Rule:**

(1) Derivative of constant function: ddx(c)=0

(2) Power Rule: ddx(xn)=nxn−1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Difference rule: ddx(f−g)=ddx(f)−ddx(g)

(5) Constant multiple rule: ddx(cf)=cddx(f)

**Calculation:**

The derivative of the function f(x) is f′(x), which is obtained as follows,

f′(x)=ddx(f(x))=ddx(x4−3x3+6x2+7x+30)

Apply the sum rule (3) and the difference rule (4),

f′(x)=ddx(x4−3x3)+ddx(6x2)+ddx(7x)+ddx(30)=ddx(x4)−ddx(3x3)+ddx(6x2)+ddx(7x)+ddx(30)

Apply the constant multiple rule (5) and the rule (1),

f′(x)=ddx(x4)−3ddx(x3)+6ddx(x2)+7ddx(x)+ddx(30)=ddx(x4)−3ddx(x3)+6ddx(x2)+7ddx(x)+0

Apply the power rule (2) and simplify the expression,

f′(x)=(4x4−1)−3(3x3−1)+6(2x2−1)+7(1x1−1)=(4x3)−3(3x2)+6(2x)+7(1x0)=4x3−9x2+12x+7

Therefore, the derivative of f(x)=x4−3x3+6x2+7x+30 is 4x3−9x2+12x+7.

**Graph:**

Use the online graphing calculator to draw the graph of the derivative function f′(x) as shown below in Figure 3.

From Figure 3, it is observed that the given polynomial function is differentiable everywhere.