
Math
Using First Derivative Test, we follow: f'(a) >0 and f'(b) <0 in the interval a<c<b implies concave down and local maximum point occurs at x=c. f'(a) <0 and f'(b) >0 in the...

Math
Given f(x)=x^312x+2: f'(x)=3x^212 and f''(x)=6x (a) The function increases when f'(x)>0 and decreases when f'(x)<0, so 3x^212>0 ==> x^2>4 ==> x<2 or x>2...

Math
Given: `f(x)=36x+3x^22x^3` Find the critical numbers by setting first derivative equal to zero and solving for the x value(s). `f'(x)=36+6x6x^2=0` `f'(x)=6+xx^2=0` `f'(x)=x^2x6=0`...

Math
Given: `f(x)=2+2x^2x^4` Find the critical values by setting the first derivative equal to zero and solving for the x values. `f'(x)=4x4x^3=0` `4x(1x^2)=0` `x=0,x=1,x=1` The critical...

Math
`g(x)=200+8x^3+x^4` differentiating, `g'(x)=24x^2+4x^3` `g'(x)=4x^2(x+6)` Now let us find our critical numbers by setting g'(x)=0 `4x^2(x+6)=0` solving above , x=0 and x=6 Now let us check sign of...

Math
`h(x)=(x+1)^55x2` differentiating, `h'(x)=5(x+1)^45` Now let us find the critical numbers, `5(x+1)^45=0` `5x(x+2)(x^2+2x+2)=0` solving above , x=0 , x=2 , x=1+i , x=1i ignore the points...

Math
Given: `h(x)=5x^33x^5` Find the critical numbers by setting the first derivative equal to zero and solving for the x values. `h'(x)=15x^215x^4=0` `15x^2(1x^2)=0` `x=0,x=1,x=1` The...

Math
`f(x)=xsqrt(6x)` differentiating, `f'(x)=xd/dxsqrt(6x)+sqrt(6x)` `f'(x)=x(1/2)(6x)^(1/2)(1)+sqrt(6x)` `f'(x)=x/(2sqrt(6x))+sqrt(6x)` `f'(x)=(x+2(6x))/(2sqrt(6x))`...

Math
a) To find the intervals of increasing or decreasing f(x), recall: > f'(x) = positive value implies increasing f(x) of an interval I. > f'(x) = negative value implies decreasing...

Math
You need to determine the monotony of the function, hence, you need to find the intervals where f'(x)>0 or f'(x)<0. You need to find the derivative of the function, using the product rule:...

Math
You need to evaluate the monotony of the function, hence, you need to remember that the function increases if f'(x)>0 and the function decreases if f'(x)<0. You need to evaluate the first...

Math
You need to determine the monotony of the function, hence, you need to find where the derivative is positive or negative. You need to evaluate the first derivative, using the chain rule: `f'(theta)...

Math
a) You need to determine the monotony of the function, hence, you need to remember that f(x) increases if f'(x)>0 and f(x) decreases if f'(x)<0. You need to evaluate f'(x), such that: `f'(x)...

Math
a) You need to determine the monotony of the function, hence, you need to verify where f'(x)>0 or f'(x)<0. Hence, you need to differentiate the function with respect to x, such that: `f'(x) =...

Math
`f(x) =x^4  2x^2 + 3` (a) To solve, take the derivative of the given function. `f'(x) =4x^3  4x` Then, set the derivative equal to zero. `0=4x^34x` Factor the right side of the equation....

Math
`f(x) = x/(x^2+1)` (a) Take the derivative of the given function. `f'(x) = ((x^2+1)(1)  (x)(2x))/(x^2+1)^2= (x^2+12x^2)/(x^2+1)^2` `f'(x)=(1x^2)/(x^2+1)^2` Then, solve for the critical numbers...

Math
`f(x)=sinx+cosx` differentiating, `f'(x)=cosxsinx` Now let us find the critical points by setting f'(x)=0, `cosxsinx=0` cosx=sinx tanx=1 x=pi/4 , 5pi/4 Now let us check the sign of f'(x) to find...

Math
a) You need to determine the intervals where the function increases and decreases, hence, you need to determine where f'(x)>0 or f'(x)<0. You need to determine the first derivative, using the...

Math
You need to determine where the function increases or decreases, hence, you need to verify where f'(x)>0 or f'(x)<0. You need to determine derivative of the function: `f'(x) = 2e^(2x) ...

Math
Rolle's Theorem has these three hypotheses: f is continuous on [1, 3], f is differentiable on (1, 3) and f(1)=f(3). The first and the second are obvious for a polynomial function, the third can be...

Math
Hypotheses 1, 2: f is contionuous on [a, b] and is differentiable on (a, b). It is obvious for a polynomial function. Hypothesis 3: f(a)=f(b), here f(a)=f(0)=2, f(b)=f(3)=27918+2=2. So the...

Math
Rolle's Theorem requires that f is continuous on closed interval (true), f is differentiable on the open interval (also true) and f(a)=f(b). f(0) = 0  0 = 0, f(9) = 3  9/3 = 0, also true. Then...

Math
Given the function `f(x)=cos(2x)` in the interval `[pi/8, (7pi)/8]` We have to see whether it sattisfies the three hypotheses of Roll's theorem. (a) f(x) is continuous in the given interval...

Math
Given: `f(x)=2x^23x+1,[0,2]` The function is a continuous polynomial on the closed interval [0,2]. The function is differentiable on the open interval (0,2). `f'(x)=4x3` `f(c)=4c3`...

Math
Given: `f(x)=x^33x+2,[2,2]` The function is continuous on the closed interval [2,2]. The function is differentiable on the open interval (2,2). `f'(x)=3x^23` `f'(c)=3c^23`...

Math
The Mean Value Theorem requires that f be continuous on [1, 4] and differentiable on (1, 4). This is true because ln(x) is differentiable for x>0. c mentioned is a number from (1, 4) such that...

Math
It is sufficient for f to be continuous on [1, 3] and differentiable on (1, 3). This is obviously true (because 0, the only gap of f, isn't in this interval). Then by the Mean Value theorem there...

Math
For the mean value theorem to be valid, the function `f(x) = sqrt x` must satisfy the following conditions on the interval, such that: f(x) is continuous over the interval [0,4] and it is because...

Math
Mean Value Theorem states a function f(x) that the satisfies the following hypotheses: 1. a function f(x) that is continuous on the closed interval [a,b] 2. differentiable on the open interval...

Math
Denote f(x)=2x+cos(x). f is continuous and diffetentiable everywhere. Also f is strictly monotone because f'(x)=2sin(x)>0. Also, f(1)=cos(1)2<0 and f(0)=1>0. By the Intermediate Value...

Math
Denote `f(x)=x^3+e^x.` f is continuous everywhere. When x tends to `+oo` , f(x) tends to `+oo` , when x tends to `oo,` f(x) tends to `oo.` In another words, there are `x_1` such that `f(x_1) lt...

Math
Hello! Denote the first number by x. Then the second will be 2*x  9 ("9 less than twice the other"). Its product p is `x*(2x9) = 2x^2  9x.` It is not difficult to find the minimum value of this...

Math
You may use binomial theorem, by itself or combined with other theorems, to solve different problems. For example, you may write the number "e", not only as a limit, but as a series, using binomial...

Math
Hello! This sounds very simple, and differentiability of f is not necessary. What does it mean by definition that "f has a maximum value on an interval I at x=c"? That for any `x in I,` `x != c`,...

Math
So in this problem the chances of someone winning the game is 6 out of every 16. This could also be written as `6/16` This is just telling us that for every 16 people who play the game, we should...

Math
Compound Interest every year: A = 600(1+0.04)^7 This produces a total amount of £789.56

Math
You need to save $1,000,000 in the next 30 years; your investment earns 5% annual interest. Your present value PV is zero, the future value FV is $1,000,000, the interest rate i is .05, and the...

Math
Hello! I hope you can use variables, "x"'s. Denote the length of the tour to Japan as J days, to Hong Kong as H and to Europe as E. What is given about them? "the tour of Japan is twice as long as...

Math
Hello! When it is said that two triangles are congruent, the order in which the vertices are listed is important. In our case, the corresponding vertices are F and T, I and O, X and P. This means...

Math
Hello! I think your function is `y(x) = 2*tan(pi*x/4).` First, a tangent line goes through the point at which it tangents. In our case this point is (1, y(1)), and `y(1) = 2tan(pi/4) = 2.`...

Math
Hello! This is a good question, but the answer is not complicated. Division is by definition operation inverse to multiplication. a/b = c by definition means that a = b*c. Now about division by...

Math
There are 27 students in the class. The maximum number of groups of 4 is 6, since 6x4=24 and 7x4=28. (There cannot be 7 groups as there aren't enough students, but there are 6 groups.) Once 6...

Math
Determine if the sequence `a_n=(1)^n9^n` converges or diverges. The first few terms are 9,81,729,6561,... This sequence is divergent since it is neither bounded below or above. (There are terms...

Math
A right cone, is a solid of revolution generated by the rotation of a right triangle, around one leg. The circle described by the other leg is called the base of the cone and the upper end of the...

Math
There is a mathematical formula which will produce the number of combinations, or permutations, from a number of selections. Note that combinations and permutations have different mathematical...

Math
Hello! This all isn't very complicated, but requires some work. Probably you know that if we take the same number (x) several times (n) and have to sum them up, then the result is x*n (x multiplied...

Math
The average weight is given as 22lbs with a population standard deviuation of 3lbs. Assuming a normal distribution of weights, what is the weight at the 70th percentile? We use a standard normal...

Math
Let's denote the number of downloads of the standard version by x and the number of downloads of the highquality version by y. Since yesterday the highquality version was downloaded four times as...

Math
Hello! In general, the answer is "yes". Unfortunately your question isn't very specific, so I'll try to guess. If you mean linear functions of one variable, they have the form y=ax+b. If another...

Math
This function is differentiable everywhere. Therefore it reaches its absolute minimum and maximum either at the endpoints or where its derivative is zero. f(2) = 16  12 + 24 + 1 = 3, f(3) = 54...