Maxima and minima; tangents and normals — practice questions

Question 1 · difficulty L2 · understanding

Let $$f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R$$. Then f :

• A. has a local minina at $$x = {1 \over 2}$$
• B. has a local minima at $$x = {3 \over 4}$$
• C. is increasing in $$\left( {{1 \over 2},{3 \over 4}} \right)$$
• D. is decreasing in $$\left( {{1 \over 2},{4 \over 3}} \right)$$

Question 2 · difficulty L2 · understanding

For which of the following curves, the line $$x + \sqrt 3 y = 2\sqrt 3$$ is the tangent at the point $$\left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \right)$$?

• A. $$2{x^2} - 18{y^2} = 9$$
• B. $${y^2} = {1 \over {6\sqrt 3 }}x$$
• C. $${x^2} + 9{y^2} = 9$$
• D. $${x^2} + {y^2} = 7$$

Question 3 · difficulty L3 · understanding

Let $\mathrm{A}(4,-2), \mathrm{B}(1,1)$ and $\mathrm{C}(9,-3)$ be the vertices of a triangle ABC . Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and $A B$ of the triangle $A B C$ respectively, is___________

Question 4 · difficulty L3 · understanding

If the set of all values of $a$, for which the equation $5 x^3-15 x-a=0$ has three distinct real roots, is the interval $(\alpha, \beta)$, then $\beta-2 \alpha$ is equal to _________.

Question 5 · difficulty L3 · understanding

Let $$\mathrm{A}$$ be the region enclosed by the parabola $$y^2=2 x$$ and the line $$x=24$$. Then the maximum area of the rectangle inscribed in the region $$\mathrm{A}$$ is ________.

Question 6 · difficulty L3 · understanding

Let $$f(x)=2^x-x^2, x \in \mathbb{R}$$. If $$m$$ and $$n$$ are respectively the number of points at which the curves $$y=f(x)$$ and $$y=f^{\prime}(x)$$ intersect the $$x$$-axis, then the value of $$\mathrm{m}+\mathrm{n}$$ is ___________.

Question 7 · difficulty L3 · understanding

Consider the triangles with vertices $A(2,1), B(0,0)$ and $C(t, 4), t \in[0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6 \alpha+21 \beta$ is equal to ___________.

Question 8 · difficulty L3 · understanding

Let $$f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R$$. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ____________.

Question 9 · difficulty L3 · understanding

Let f(x) be a cubic polynomial with f(1) = $$-$$10, f($$-$$1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $$-$$1. Then f(3) is equal to ____________.

Question 10 · difficulty L3 · understanding

The number of distinct real roots of the equation 3x 4 + 4x 3 $$-$$ 12x 2 + 4 = 0 is _____________.