DP Mathematics HL Questionbank

• Syllabus

6.3

Sub sections and their related questions

Local maximum and minimum values.

• 12M.2.hl.TZ1.10: A triangle is formed by the three lines \(y = 10 - 2x,{\text{ }}y = mx\) and...
• 12N.1.hl.TZ0.4a: Find the coordinates of the points A and B.
• 12N.2.hl.TZ0.12b: If a = 5 and b = 1, find the maximum length of a painting that can be removed through this doorway.
• 12N.2.hl.TZ0.12d: Let a = 3k and b = k .  Find, in terms of k , the maximum length of a painting that can be...
• 12N.2.hl.TZ0.12e: Let a = 3k and b = k .  Find the minimum value of k if a painting 8 metres long is to be removed...
• 08M.1.hl.TZ1.5: If \(f(x) = x - 3{x^{\frac{2}{3}}},{\text{ }}x > 0\) , (a)     find the x-coordinate of the...
• 08M.1.hl.TZ1.12: The function f is defined by \(f(x) = x{{\text{e}}^{2x}}\) . It can be shown that...
• 11M.1.hl.TZ2.1a: Find the value of p and the value of q .
• 11M.1.hl.TZ2.11a: Find the coordinates of the points on C at which \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\) .
• 13M.2.hl.TZ1.7b: Find the value of x, to the nearest metre, such that this cost is minimized.
• 10M.1.hl.TZ1.11: Consider \(f(x) = \frac{{{x^2} - 5x + 4}}{{{x^2} + 5x + 4}}\). (a)     Find the equations of all...
• 10M.1.hl.TZ2.7: The function f is defined by \(f(x) = {{\text{e}}^{{x^2} - 2x - 1.5}}\). (a)     Find...
• 10M.2.hl.TZ2.11: The function f is defined...
• 10N.2.hl.TZ0.13: Let \(f(x) = \frac{{a + b{{\text{e}}^x}}}{{a{{\text{e}}^x} + b}}\), where \(0 < b <...
• 13M.2.hl.TZ2.13e: Using the result in part (d), or otherwise, determine the value of x corresponding to the maximum...
• 11M.1.hl.TZ1.12a: (i)     Solve the equation \(f'(x) = 0\) . (ii)     Hence show the graph of \(f\) has a local...
• 11M.2.hl.TZ1.2a: Find the equation of the straight line passing through the maximum and minimum points of the...
• 09N.2.hl.TZ0.3: (a)     Show that the area of the shaded region is \(8\sin x - 2x\) . (b)     Find the maximum...
• 09M.2.hl.TZ2.7: (a)     Show that \({b^2} > 24c\) . (b)     Given that the coordinates of P and Q are...
• 09M.2.hl.TZ2.13: (a)     On the same set of axes draw, on graph paper, the graphs, for...
• 14M.1.hl.TZ1.11c: Find the coordinates of C, the point of inflexion on the curve.
• 14M.1.hl.TZ2.13d: Find the \(x\)-coordinates of the other two points of inflexion.
• 13N.1.hl.TZ0.10c: Find the coordinates of B, the point of inflexion.
• 14M.1.hl.TZ1.11b: Find the coordinates of B, at which the curve reaches its maximum value.
• 14M.1.hl.TZ2.13b: Hence find the \(x\)-coordinates of the points where the gradient of the graph of \(f\) is zero.
• 13N.1.hl.TZ0.10b: Find an expression for \(f''(x)\) and hence show the point A is a maximum.
• 14N.1.hl.TZ0.5: A tranquilizer is injected into a muscle from which it enters the bloodstream. The concentration...
• 14N.2.hl.TZ0.10c: (i)     Find \(\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}}\) and hence justify that...
• 15M.1.hl.TZ1.11c: Find the coordinates of any local maximum and minimum points on the graph of \(y(x)\). Justify...
• 15M.1.hl.TZ2.6b: Given that \(AB\) has a minimum value, determine the value of \(\theta \) for which this occurs.
• 15N.1.hl.TZ0.12c: Hence find the \(x\)-coordinates of any local maximum or minimum points.
• 15N.2.hl.TZ0.13b: Show that the gradient of the roof function is greatest when \(x =  - \sqrt {200} \).
• 16M.2.hl.TZ1.11b: For the curve \(y = f(x)\). (i)     Find the coordinates of both local minimum points. (ii)    ...
• 16N.1.hl.TZ0.11c: Show that the function \(f\) has a local maximum value when \(x = \frac{{3\pi }}{4}\).
• 18M.1.hl.TZ1.9a: The graph of \(y = f\left( x \right)\) has a local maximum at A. Find the coordinates of A.
• 18M.2.hl.TZ1.9a: Show that there are exactly two points on the curve where the gradient is zero.
• 18M.1.hl.TZ2.4: Consider the curve \(y = \frac{1}{{1 - x}} + \frac{4}{{x - 4}}\). Find the x-coordinates of the...

Optimization problems.

• 08M.1.hl.TZ2.13: André wants to get from point A located in the sea to point Y located on a straight stretch of...
• 08N.1.hl.TZ0.9: A packaging company makes boxes for chocolates. An example of a box is shown below. This box is...
• 13M.2.hl.TZ1.7b: Find the value of x, to the nearest metre, such that this cost is minimized.
• 11N.1.hl.TZ0.8c: What route should Jorg take to travel from A to B in the least amount of time? Give reasons for...
• 11N.1.hl.TZ0.11a: Determine the time at which the two ships are closest to one another, and justify your answer.
• 11N.1.hl.TZ0.11b: If the visibility at sea is 9 km, determine whether or not the captains of the two ships can ever...
• 09M.2.hl.TZ2.12: (a)     Explain why \(x < 40\) . (b)     Show that cosθ = x −10 50. (c)     (i)     Find an...
• 14M.1.hl.TZ1.11c: Find the coordinates of C, the point of inflexion on the curve.
• 14M.1.hl.TZ2.13d: Find the \(x\)-coordinates of the other two points of inflexion.
• 14M.2.hl.TZ2.12: Engineers need to lay pipes to connect two cities A and B that are separated by a river of width...
• 14M.1.hl.TZ1.11b: Find the coordinates of B, at which the curve reaches its maximum value.
• 14M.1.hl.TZ2.13b: Hence find the \(x\)-coordinates of the points where the gradient of the graph of \(f\) is zero.
• 15N.2.hl.TZ0.13c: The cross section of the living space under the roof can be modelled by a rectangle \(CDEF\) with...
• 16M.2.hl.TZ2.11c: (i)     Find \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha )\). (ii)     Hence or otherwise...
• 16M.2.hl.TZ2.11d: Find the set of values of \(x\) for which \(\alpha  \geqslant 7^\circ \).
• 17M.1.hl.TZ2.10a.i: Find the area of the window in terms of P and \(r\).
• 17M.1.hl.TZ2.10a.ii: Find the width of the window in terms of P when the area is a maximum, justifying that this is a...
• 17M.1.hl.TZ2.10b: Show that in this case the height of the rectangle is equal to the radius of the semicircle.

Points of inflexion with zero and non-zero gradients.

• 12M.1.hl.TZ1.12b: Show that the curve \(y = f(x)\) has one point of inflexion, and find its coordinates.
• 12N.1.hl.TZ0.4b: Given that the graph of the function has exactly one point of inflexion, find its coordinates.
• 08M.1.hl.TZ1.12: The function f is defined by \(f(x) = x{{\text{e}}^{2x}}\) . It can be shown that...
• 08M.2.hl.TZ1.13: A family of cubic functions is defined as...
• 08M.2.hl.TZ2.6: Consider the curve with equation \(f(x) = {{\text{e}}^{ - 2{x^2}}}{\text{ for }}x < 0\)...
• 08N.2.hl.TZ0.12: The function f is defined by...
• SPNone.1.hl.TZ0.5c: John states that, because \(f''(0) = 0\) , the graph of f has a point of inflexion at the point...
• SPNone.2.hl.TZ0.13c: Find the x-coordinates of the two points of inflexion on the graph of f .
• 10M.1.hl.TZ1.11: Consider \(f(x) = \frac{{{x^2} - 5x + 4}}{{{x^2} + 5x + 4}}\). (a)     Find the equations of all...
• 10N.2.hl.TZ0.13: Let \(f(x) = \frac{{a + b{{\text{e}}^x}}}{{a{{\text{e}}^x} + b}}\), where \(0 < b <...
• 11M.1.hl.TZ1.12b: Show that there is a point of inflexion on the graph and determine its coordinates.
• 11M.2.hl.TZ1.2b: Show that the point of inflexion of the graph \(y = f (x)\) lies on this straight line.  
• 14M.1.hl.TZ1.11c: Find the coordinates of C, the point of inflexion on the curve.
• 14M.1.hl.TZ2.13d: Find the \(x\)-coordinates of the other two points of inflexion.
• 14M.1.hl.TZ1.11b: Find the coordinates of B, at which the curve reaches its maximum value.
• 14M.1.hl.TZ2.13b: Hence find the \(x\)-coordinates of the points where the gradient of the graph of \(f\) is zero.
• 15M.1.hl.TZ1.11d: Find the coordinates of any points of inflexion on the graph of \(y(x)\). Justify whether any...
• 15M.1.hl.TZ2.4b: There is a point of inflexion, \(P\), on the curve \(y = f(x)\). Find the coordinates of \(P\).
• 16M.2.hl.TZ1.11b: For the curve \(y = f(x)\). (i)     Find the coordinates of both local minimum points. (ii)    ...
• 16N.1.hl.TZ0.11d: Find the \(x\)-coordinate of the point of inflexion of the graph of \(f\).
• 16N.1.hl.TZ0.11e: Sketch the graph of \(f\), clearly indicating the position of the local maximum point, the point...
• 18M.1.hl.TZ1.9b.i: Show that there is exactly one point of inflexion, B, on the graph of \(y = f\left( x \right)\).
• 18M.1.hl.TZ1.9b.ii: The coordinates of B can be expressed in the form...

Graphical behaviour of functions, including the relationship between the graphs of \(f\) , \({f'}\) and \({f''}\) .

• 12M.1.hl.TZ1.2b: Hence state the value of (i)     \(f'( - 3)\); (ii)     \(f'(2.7)\); (iii)  ...
• 12M.1.hl.TZ2.7b: On the axes below, sketch the graph of the derivative \(y = f'(x)\) , clearly showing the...
• 09M.1.hl.TZ1.11: Let f be a function defined by \(f(x) = x - \arctan x\) , \(x \in \mathbb{R}\) . (a)     Find...
• 09N.1.hl.TZ0.9: The diagram below shows a sketch of the gradient function \(f'(x)\) of the curve \(f(x)\)...
• SPNone.2.hl.TZ0.13b: Sketch the graphs of f and \(f'\) on the same axes, showing clearly all x-intercepts.
• 10M.1.hl.TZ1.11: Consider \(f(x) = \frac{{{x^2} - 5x + 4}}{{{x^2} + 5x + 4}}\). (a)     Find the equations of all...
• 11M.1.hl.TZ1.12c: Sketch the graph of \(y = f(x)\) , indicating clearly the asymptote, x-intercept and the local...
• 09N.2.hl.TZ0.10: (a)     Find in terms of \(a\)   (i)     the zeros of \(f\) ;   (ii)     the values of \(x\)...
• 09M.2.hl.TZ2.13: (a)     On the same set of axes draw, on graph paper, the graphs, for...