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Q3(viii):
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
(viii) $f(x) = x \sqrt{1-x}, 0 < x < 1$
Solution :
Step 1: Establishing the Function and Domain
Given function: $f(x) = x\sqrt{1-x}$ defined on the open interval $(0, 1)$.
To determine local extrema, we first calculate the first derivative $f'(x)$ with respect to $x$ using the product rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$.
Step 2: First-Order Differentiation
$f'(x) = \frac{d}{dx}[x] \cdot \sqrt{1-x} + x \cdot \frac{d}{dx}[\sqrt{1-x}]$
$f'(x) = 1 \cdot \sqrt{1-x} + x \cdot \left( \frac{1}{2\sqrt{1-x}} \cdot (-1) \right)$ [By Chain Rule]
$f'(x) = \sqrt{1-x} - \frac{x}{2\sqrt{1-x}}$
Finding a common denominator:
$f'(x) = \frac{2(1-x) - x}{2\sqrt{1-x}} = \frac{2 - 2x - x}{2\sqrt{1-x}} = \frac{2 - 3x}{2\sqrt{1-x}}$
Step 3: Critical Point Analysis
Set $f'(x) = 0$ to identify critical points within the interval $(0, 1)$:
$\frac{2 - 3x}{2\sqrt{1-x}} = 0 \implies 2 - 3x = 0 \implies x = \frac{2}{3}$
Since $\frac{2}{3} \in (0, 1)$, it is a valid critical point.
Step 4: Second-Order Differentiation for Concavity
To apply the Second Derivative Test, we differentiate $f'(x) = \frac{1}{2} (2-3x)(1-x)^{-1/2}$:
$f''(x) = \frac{1}{2} \left[ -3(1-x)^{-1/2} + (2-3x) \cdot \left( -\frac{1}{2}(1-x)^{-3/2} \cdot (-1) \right) \right]$
$f''(x) = \frac{1}{2} \left[ \frac{-3}{\sqrt{1-x}} + \frac{2-3x}{2(1-x)^{3/2}} \right]$
$f''(x) = \frac{1}{2} \left[ \frac{-6(1-x) + 2 - 3x}{2(1-x)^{3/2}} \right] = \frac{-6 + 6x + 2 - 3x}{4(1-x)^{3/2}} = \frac{3x - 4}{4(1-x)^{3/2}}$
Step 5: Applying the Second Derivative Test
Evaluate $f''(x)$ at $x = \frac{2}{3}$:
$f''\left(\frac{2}{3}\right) = \frac{3(\frac{2}{3}) - 4}{4(1-\frac{2}{3})^{3/2}} = \frac{2 - 4}{4(1/3)^{3/2}} = \frac{-2}{4(1/3\sqrt{3})} = \frac{-1}{2(1/3\sqrt{3})} = -\frac{3\sqrt{3}}{2}$
Since $f''(\frac{2}{3}) < 0$, the function attains a local maximum at $x = \frac{2}{3}$ [Per Second Derivative Test].
Step 6: Calculation of Local Maximum Value
$f\left(\frac{2}{3}\right) = \frac{2}{3} \sqrt{1 - \frac{2}{3}} = \frac{2}{3} \sqrt{\frac{1}{3}} = \frac{2}{3\sqrt{3}} = \frac{2\sqrt{3}}{9}$
Final Solution: The function has a local maximum at $x = \frac{2}{3}$ with a local maximum value of $\frac{2\sqrt{3}}{9}$. There is no local minimum.
More Questions from Class 12 Mathematics Application of Derivatives EXERCISE 6.3
- Q1(i): Find the maximum and minimum values, if any, of the following functions given by (i) $f(x) = (2x – 1)^2 + 3$
- Q1(ii): Find the maximum and minimum values, if any, of the following functions given by (ii) $f(x) = 9x^2 + 12x + 2$
- Q1(iii): Find the maximum and minimum values, if any, of the following functions given by (iii) $f(x) = – (x – 1)^2 + 10$
- Q1(iv): Find the maximum and minimum values, if any, of the following functions given by (iv) $g(x) = x^3 + 1$
- Q2(i): Find the maximum and minimum values, if any, of the following functions given by (i) $f(x) = | x + 2 | – 1$
- Q2(ii): Find the maximum and minimum values, if any, of the following functions given by (ii) $g(x) = – | x + 1| + 3$
- Q2(iii): Find the maximum and minimum values, if any, of the following functions given by (iii) $h(x) = \sin(2x) + 5$
- Q2(iv): Find the maximum and minimum values, if any, of the following functions given by (iv) $f(x) = | \sin 4x + 3|$
- Q2(v): Find the maximum and minimum values, if any, of the following functions given by (v) $h(x) = x + 1, x \in (– 1, 1)$
- Q3(i): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i) $f(x) = x^2$
- Q3(ii): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (ii) $g(x) = x^3 – 3x$
- Q3(iii): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (iii) $h(x) = \sin x + \cos x, 0 < x < \frac{\pi}{2}$
- Q3(iv): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (iv) $f(x) = \sin x – \cos x, 0 < x < 2\pi$
- Q3(v): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (v) $f(x) = x^3 – 6x^2 + 9x + 15$
- Q3(vi): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (vi) $g(x) = \frac{x}{2} + \frac{2}{x}, x > 0$
- Q3(vii): Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (vii) $g(x) = \frac{1}{x^2 + 2}$
- Q4(i): Prove that the following functions do not have maxima or minima: (i) $f(x) = e^x$
- Q4(ii): Prove that the following functions do not have maxima or minima: (ii) $g(x) = \log x$
- Q4(iii): Prove that the following functions do not have maxima or minima: (iii) $h(x) = x^3 + x^2 + x +1$
- Q5(i): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) $f(x) = x^3, x \in [– 2, 2]$
- Q5(ii): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (ii) $f(x) = \sin x + \cos x, x \in [0, \pi]$
- Q5(iii): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iii) $f(x) = 4x - \frac{1}{2}x^2, x \in [-2, \frac{9}{2}]$
- Q5(iv): Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iv) $f(x) = (x-1)^2 + 3, x \in [-3, 1]$
- Q6: Find the maximum profit that a company can make, if the profit function is given by $p(x) = 41 – 72x – 18x^2$
- Q7: Find both the maximum value and the minimum value of $3x^4 – 8x^3 + 12x^2 – 48x + 25$ on the interval $[0, 3]$.
- Q8: At what points in the interval $[0, 2\pi]$, does the function $\sin 2x$ attain its maximum value?
- Q9: What is the maximum value of the function $\sin x + \cos x$?
- Q10: Find the maximum value of $2x^3 – 24x + 107$ in the interval $[1, 3]$. Find the maximum value of the same function in $[–3, –1]$.
- Q11: It is given that at $x = 1$, the function $x^4 – 62x^2 + ax + 9$ attains its maximum value, on the interval $[0, 2]$. Find the value of $a$.
- Q12: Find the maximum and minimum values of $x + \sin 2x$ on $[0, 2\pi]$.
- Q13: Find two numbers whose sum is $24$ and whose product is as large as possible.
- Q14: Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum.
- Q15: Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^2 y^5$ is a maximum.
- Q16: Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.
- Q17: A square piece of tin of side $18$ cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.
- Q18: A rectangular sheet of tin $45$ cm by $24$ cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?
- Q19: Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
- Q20: Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
- Q21: Of all the closed cylindrical cans (right circular), of a given volume of $100$ cubic centimetres, find the dimensions of the can which has the minimum surface area?
- Q22: A wire of length $28$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
- Q23: Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $\frac{8}{27}$ of the volume of the sphere.
- Q24: Show that the right circular cone of least curved surface and given volume has an altitude equal to $\sqrt{2}$ time the radius of the base.
- Q25: Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is $\tan^{-1} \sqrt{2}$.
- Q26: Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $\sin^{-1}(\frac{1}{3})$.
- Q27: The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is
- Q28: For all real values of $x$, the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is
- Q29: The maximum value of $[x(x-1)+1]^{\frac{1}{3}}$, $0 \le x \le 1$ is
CBSE Solutions for Class 12 Mathematics Application of Derivatives
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