$\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-2)=-17\\&f(1)=-4\\&f(4)=-16\\&f(7)=-16\\&\text{Although the total interval has a length of }9\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }3\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-2}^{7}f(x)\approx3[(-17)+(-4)+(-16)]\\&\int_{-2}^{7}f(x)\approx-111\end{align*}$

$\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(5)=-3\\&f(11)=8\\&f(17)=11\\&f(23)=-3\\&\text{Although the total interval has a length of }18\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }6\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{5}^{23}f(x)\approx6[(-3)+(8)+(11)]\\&\int_{5}^{23}f(x)\approx96\end{align*}$

$\displaystyle \begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-8)=-10\\&f(1)=12\\&f(10)=-1\\&f(19)=11\\&\text{Although the total interval has a length of }27\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }9\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-8}^{19}f(x)\approx9[(-10)+(12)+(-1)]\\&\int_{-8}^{19}f(x)\approx9\end{align*}$

At a relative minimum $\displaystyle (c,f(c))$ of the graph $\displaystyle f(x)$, it will hold that $\displaystyle f'(c) = 0$ and $\displaystyle f''(c) < 0$. 

First, find $\displaystyle f'(x)$. Using the sum rule,

$\displaystyle f'(x) = \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{1}{7}x^{7} - \frac{6}{5} x^{5} + \frac{5}{3}x^{3} - 30x \right )$

$\displaystyle f'(x) = \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{1}{7}x^{7} \right )- \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{6}{5} x^{5} \right )+ \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{5}{3}x^{3} \right )- \frac{\mathrm{d} }{\mathrm{d} x}\left ( 30x \right )$

Differentiate the individual terms using the Constant Multiple and Power Rules:

$\displaystyle f'(x) = \frac{1}{7} \cdot 7 x^{6} - \frac{6}{5} \cdot 5 x^{4} + \frac{5}{3} \cdot 3 x^{2} - 30x$

$\displaystyle f'(x) = x^{6} - 6 x^{4} +5 x^{2} - 30$

Set this equal to 0:

$\displaystyle x^{6} - 6 x^{4} +5 x^{2} - 30 = 0$

$\displaystyle (x^{6} - 6 x^{4} )+(5 x^{2} - 30) = 0$

$\displaystyle x^{4} (x^{2} - 6 )+ 5 (x^{2} - 6 ) = 0$

$\displaystyle (x^{4}+ 5 ) (x^{2} - 6 ) = 0$

Either:

$\displaystyle x^{4}+5 = 0$, in which case, $\displaystyle x^{4} = -5$; this equation has no real solutions.

$\displaystyle x^{2}- 6=0$ has two real solutions, $\displaystyle -\sqrt {6}$ and $\displaystyle \sqrt {6}$. 

Now take the second derivative, again using the sum rule:

$\displaystyle f'(x) = x^{6} - 6 x^{4} +5 x^{2} - 30$

$\displaystyle f''(x) = \frac{\mathrm{d} }{\mathrm{d} x}(x^{6} - 6 x^{4} +5 x^{2} - 30)$

$\displaystyle f''(x) = \frac{\mathrm{d} }{\mathrm{d} x}(x^{6} )- \frac{\mathrm{d} }{\mathrm{d} x} ( 6 x^{4} )+\frac{\mathrm{d} }{\mathrm{d} x}( 5 x^{2} )- \frac{\mathrm{d} }{\mathrm{d} x}( 30)$

Differentiate the individual terms using the Constant Multiple and Power Rules:

$\displaystyle f''(x) =6 x^{5} - 6 ( 4 x^{3} )+ 5( 2x )- 0$

$\displaystyle f''(x) =6 x^{5} -2 4 x^{3} +10x$

Substitute $\displaystyle \sqrt {6}$ for $\displaystyle x$:

$\displaystyle f''( \sqrt {6}) =6 ( \sqrt {6} )^{5} -2 4 ( \sqrt {6})^{3} +10( \sqrt {6})$

$\displaystyle f''(-\sqrt {6}) = 216\sqrt {6}-144 \sqrt {6} +10\sqrt {6}$

$\displaystyle f''(-\sqrt {6}) =82\sqrt {6} >0$

Therefore, $\displaystyle f$ has a relative minimum at $\displaystyle x = \sqrt {6}$.

Now. substitute $\displaystyle -\sqrt {6}$ for $\displaystyle x$:

$\displaystyle f''(-\sqrt {6}) =6 (-\sqrt {6} )^{5} -2 4 (-\sqrt {6})^{3} +10(-\sqrt {6})$

$\displaystyle f''(-\sqrt {6}) =- 216\sqrt {6}+144 \sqrt {6} -10\sqrt {6}$

$\displaystyle f''(-\sqrt {6}) =-82\sqrt {6} < 0$

Therefore, $\displaystyle f$ has a relative maximum at $\displaystyle x =- \sqrt {6}$.

Because our $\displaystyle \Delta x$ is constant, the left Riemann sum will be

$\displaystyle R_{L}=\Delta x(f(0)+f(0.5)+f(1)+f(1.5)+f(2)+f(2.5))$

$\displaystyle R_{L}=0.5\cdot (1+1.75+4+7.75+13+19.75)=23.625$

To use left Riemann sums, we need to use the following formula:

$\displaystyle Area \approx \sum_{i=1}^n f(x_i)\Delta x$.

where $\displaystyle n$ is the number of subintervals, (4 in our problem),

$\displaystyle i$ is the "counter" that denotes which subinterval we are working with,(4 subintervals mean that $\displaystyle i$ will be 1, 2, 3, and then 4)

$\displaystyle f(x_i)$ is the function value when you plug in the "i-th" x value, (i-th in this case will be 1-st, 2-nd, 3-rd, and 4-th)

,$\displaystyle \Delta x$ is the width of each subinterval, which we will determine shortly.

and $\displaystyle \sum_{i=1}^n$ means add all $\displaystyle n$ versions together (for us that means add up 4 versions).

This fancy equation approximates using boxes. We can rewrite this fancy equation by writing $\displaystyle f(x_i)\Delta x$, 4 times; 1 time each for $\displaystyle i=1$, $\displaystyle i=2$, $\displaystyle i=3$, and $\displaystyle i=4$. This gives us

$\displaystyle f(x_1)\Delta x + f(x_2)\Delta x + f(x_3)\Delta x + f(x_4)\Delta x$

Think of $\displaystyle \Delta x$ as the base of each box, and $\displaystyle f(x_1)$ as the height of the 1st box.

This is basically $\displaystyle Area = (base)(height)$, 4 times, and then added together.

Now we need to determine what $\displaystyle \Delta x, x_1, x_2, x_3,$ and $\displaystyle x_4$ are.

To find $\displaystyle \Delta x$ we find the total length between the beginning and ending x values, which are given in the problem as $\displaystyle x=-2$ and $\displaystyle x=2$. We then split this total length into 4 pieces, since we are told to use 4 subintervals.

In short, $\displaystyle \Delta x = \frac{b - a}{n}= \frac{(2)- (-2)}{4} = 1$

Now we need to find the x values that are the left endpoints of each of the 4 subintervals. Left endpoints because we are doing Left Riemann sums.

The left most x value happens to be the smaller of the overall endpoints given in the question. In other words, since we only care about the area from $\displaystyle x=-2$ to $\displaystyle x=2$, we'll just use the smaller one, $\displaystyle -2$, for our first $\displaystyle x_i$.

Now we know that $\displaystyle x_1 = -2$.

To find the next endpoint, $\displaystyle x_2$, just increase the first x by the length of the subinterval, which is $\displaystyle \Delta x = 1$. In other words

$\displaystyle x_2 = x_1 + \Delta x$

$\displaystyle x_2 = -2 + 1$

$\displaystyle x_2 = -1$

Add the $\displaystyle \Delta x$ again to get $\displaystyle x_3$

$\displaystyle x_3 = x_2 + \Delta x$

$\displaystyle x_3 = -1 + 1$

$\displaystyle x_3 = 0$

And repeat to find $\displaystyle x_4$

$\displaystyle x_4 = x_3 + \Delta x$

$\displaystyle x_4 = 0 + 1$

$\displaystyle x_4 = 1$

Now that we have all the pieces, we can plug them in.

$\displaystyle f(x_1)\Delta x + f(x_2)\Delta x + f(x_3)\Delta x + f(x_4)\Delta x$

$\displaystyle f(-2)\cdot1 + f(-1) \cdot (1)+ f(0)\cdot (1) + f(1)\cdot (1)$

plug each value into $\displaystyle f(x)=x^4 +1$ and then simplify.

$\displaystyle [(-2)^4+1](1)+[(-1)^4+1](1)+ [(0)^4+1](1)+[(1)^4+1](1)$

$\displaystyle [16 + 1]+ [1 + 1]+ [0+ 1]+ [1 + 1]$

$\displaystyle 17 + 2 + 1 + 2$

$\displaystyle 22$

This is the final answer, which is an approximation of the area under the function.

We will find the points of inflection by setting our second derivative to zero.

$\displaystyle f'(x)=\frac{1}{3}x^3+\frac{3}{2}x^2-4x+6$

Take the derivative again,

$\displaystyle f''(x)=x^2+3x-4$

then set this equal to zero and reverse foil,

$\displaystyle 0=(x-1)(x+4)$

$\displaystyle 0=x-1 \ or \ 0=x+4$

$\displaystyle x=1 \or \ x=-4$

These are our points of inflection.

To find the inflection points of $\displaystyle f(x)$, we need to find $\displaystyle f''(x)$ (which lucky for us, is already given!) set it equal to $\displaystyle 0$, and solve for $\displaystyle x$.

$\displaystyle 0=\sin(x)e^x$. Start

$\displaystyle 0=\sin(x)$. Divide by $\displaystyle e^x$. We can do this, because $\displaystyle e^x$ is never equal to $\displaystyle 0$.

On the unit circle, the values $\displaystyle 0, \pi, 2\pi$ cause $\displaystyle \sin(x)=0$, but only $\displaystyle \pi$ is inside our interval $\displaystyle (0,2\pi)$. so $\displaystyle x=\pi$ is the only value to consider here.

To prove that $\displaystyle x = \pi$ is actually part of a point of inflection, we have to test an $\displaystyle x$ value on the left and the right of $\displaystyle \pi$, and substitute them into $\displaystyle f''(x)$ and test their signs.

$\displaystyle f''(\pi/2) = \sin(\pi/2)e^{(\pi/2)} = e^{(\pi/2)} >0$

$\displaystyle f''(3\pi/2) = \sin(3\pi/2)e^{(3\pi/2)} = -e^{(3\pi/2)}< 0$.

Hence $\displaystyle x=\pi$, is the $\displaystyle x$ coordinate of an inflection point of $\displaystyle f(x)$ on the interval $\displaystyle (0,2\pi)$.

Inflection points of a function occur when the second derivative equals zero. Therefore, we simply need to take two derivatives of our function, and solve.

$\displaystyle f(x)=2x^4+4x^3$

$\displaystyle f'(x)=8x^3+12x^2$

$\displaystyle f''(x)=24x^2+24x=0=24x(x-1).$

Therefore, the two values that makes this function go to zero are $\displaystyle x=0,1.$

To determine the points of inflection, we must find the value at which the second derivative of the function changes in sign.

First, we find the second derivative:

$\displaystyle f'=x^2+4x$

$\displaystyle f''=2x+4$

The derivatives were found using the following rules:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Next, we find the values at which the second derivative of the function is equal to zero:

$\displaystyle 2x+4=0$

$\displaystyle x=-2$

Using the critical value, we now create intervals over which to evaluate the sign of the second derivative:

Notice how at the bounds of the intervals, the second derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, while on the second interval, the second derivative is positive. The second derivative changed sign $\displaystyle x=-2$, so there exists the point of inflection for the function.