DERIVATIVE OF ALGEBRAIC FUNCTIONS — Part 2 (Problems 41-73)
33 multiple-choice questions on derivatives of algebraic functions.
Try to solve each question independently before checking the answer key.
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Problem 41
Diberikan grafik fungsi $f \left( x \right)= 3x^{\frac{5}{3}}-15x^{\frac{2}{3}}$, maka...
(1) $f'(0)$ does not exist
(2) The function is increasing on $(2, \infty)$
(3) The function is decreasing on $(0, 2)$
(4) There is a relative minimum at $(2, -9\sqrt[3]{4})$
Problem 42
If the quadratic equation $x^{2}-px+q=0$ has reciprocal roots that are negative numbers, the maximum value of $p-q$ is...
Problem 43
A box with a square base is designed to have a volume of $2$ liters. The cost of making the top and bottom is $2$ thousand rupiah per $dm^{2}$ and the cost of making the side walls is $1$ thousand rupiah per $dm^{2}$. The minimum cost is $p$ thousand rupiah, where $p=\cdots$
Problem 44
For $x \geq 1$, the maximum value of the function $f(x)=-x^{3}+6x^{2}-9x+7$ is...
Problem 45
Given $f$ is a quadratic function that has a tangent line $y=-x+1$ at $x=-1$. If $f'(1)=3$, then $f(4)=\cdots$
Problem 46
Given the graph of $f'(x)$ as shown. The properties of $f(x)$ are...
(1) For $x < -1$, $f(x)$ is decreasing
(2) For $0 < x < 2$, $f(x)$ is increasing
(3) The tangent line to $f(x)$ at $x=-1$ is parallel to the $x$-axis
(4) $x=2$ is an extremum point
Problem 47
Given $f'(x) = x^{3}(x - a)^{2}(x - b)$ with $0 < a < b$.
The correct statement about $f$ is...
(1) For $x < b$, $f(a)$ is the maximum value of $f$
(2) For $x > 0$, $f(b)$ is the minimum value of $f$
(3) For $x < 0$, $f$ is decreasing
(4) For $x > b$, $f$ is increasing
Problem 48
Given the graph of $f(x)=ax^{3}+bx^{2}+cx+d$ with $b^{2} < 3ac$. The following statements may occur for $f$, except...
Problem 49
The graph of $y =\dfrac{1}{3}x^{3}-\dfrac{3}{2}x^{2}+2x$ has horizontal tangent lines at points $P$ and $Q$. The sum of the ordinates of points $P$ and $Q$ is...
Problem 50
The area of a circle is a function of its circumference. If the circumference of a circle is $x$, then the rate of change of the area with respect to its circumference is...
Problem 51
A construction project for a building can be completed in $x$ days, with a daily cost of $\left( 3x-900+\dfrac{200}{x} \right)$ hundred thousand rupiah. To minimize the total cost, the project should be completed in...
Problem 52
The shortest distance from the curve $y=\frac{1}{2}x^{2}+1$ to the line $2x-y=4$ is...
Problem 53
From a square cardboard with side length $30\ cm$, an open box is made by cutting four squares from each corner. The maximum volume of the box that can be made is...
Problem 54
An aquarium in the shape of an open rectangular prism has a base with a width-to-length ratio of $2:3$. If the surface area of the aquarium is $1,800\ \text{cm}^{2}$, the maximum volume of the aquarium is...$\text{cm}^{3}$
Problem 55
A wire of length $128\ cm$ is to be bent into five rectangles as shown in the figure. The maximum area that can be formed is...$\text{cm}^{2}$
Problem 56
The tangent lines to the curve $y=3-x^{2}$ at points $P(-a,b)$ and $Q(a,b)$ intersect the $y$-axis at point $R$. The value of $a$ that makes triangle $PQR$ equilateral is...
Problem 57
The sum of two positive numbers is $120$. To maximize the product of the square of the first number and the second number, the difference between the largest and smallest numbers is...
Problem 58
To produce $x$ units of goods per day, the cost is $\left( x^{3}-3,000x^{2}+3,000,000x \right)$ rupiah. If the goods must be produced, the minimum cost per unit is achieved when producing...per day.
Problem 59
The sum of the first number and the square of the second number is $75$. The maximum value of the product of the two numbers is...
Problem 60
If $\bigtriangleup ABC$ is an isosceles right triangle with $AC=BC=8$ and $AD=CE$, then the minimum area of quadrilateral $ABED$ is...
Problem 61
A gutter is to be made from a $30\ cm$ wide sheet of metal by folding it into three equal parts as shown. If $\theta$ is the angle between the side walls and the base $(0 < \theta < \frac{\pi}{2})$, the water capacity is maximized when $\theta=\cdots$
Problem 62
If the graph of $y=x+\dfrac{1}{x}$ attains its maximum at point $(x_{0},y_{0})$, then $x_{0}+y_{0} =\cdots$
Problem 63
A water tank is made from steel plate in the shape of a half cylinder as shown. The top is open and the capacity is $125\pi$ liters. To minimize the material used, $h=\cdots$ meters.
Problem 64
An aquarium has a rectangular base and sides, with no top. The volume is $4\ m^{3}$. The width of the base is $1\ m$. The cost of the base is $Rp10,000.00$ per $m^{2}$, and the sides cost $Rp5,000.00$ per $m^{2}$. The minimum cost to build the aquarium is...
Problem 65
A particle moves along the $t$-axis with position $s(t)=2t^{3}-24t^{2}+90t+7$, $t \geq 0$. The velocity of the particle is positive when $t$ satisfies...
Problem 66
Given $f(x)=\sqrt{x^{2}-ax+b}$. If $f(1)=f'(1)=2$, then $a+b=\cdots$
Problem 67
Given positive numbers $m$ and $n$. If $mx+ny=1$, then the maximum value of $xy$ is...
Problem 68
If the tangent line to the curve $y = x^{3} - 3x^{2} - 9x$ at point $(a,b)$ has gradient $15$, then the possible value of $a+b$ is...
Problem 69
Given functions $f$ and $g$ with $f(x)=(2x+1)^{5}$ and $h=f \circ g$. If $g(5)=-1$ and $g'\left(\frac{x+1}{x-1}\right)=2x+2$, then $h'(5)=\cdots$
Problem 70
Given $f(x)=(ax^{2}+bx+c)(x^{2}+x)$. If $f'(0)=3$ and $f'(-1)=10$, then $f'\left(-\frac{1}{2}\right)=\cdots$
Problem 71
If $m$ and $M$ respectively denote the relative minimum and maximum values of $f(x)=2x^{3}-3x^{2}+a$ with $M+m=3$, then $f(2)=\cdots$
Problem 72
If the graph below is $y=\dfrac{df(x)}{dx}$, then it can be concluded that $f(x)$...
Problem 73
Rectangle $PQRS$ lies inside right triangle $PTU$. If $PS=4$ and $PQ=3$, then the minimum area of $\bigtriangleup PTU$ is...