Q.240. MCQs of the applications of derivatives

Question 240.
NOTE :- f ' (x) indicates the 1st order derivative of f(x),
f ' ' (x) indicates 2nd order derivative of f(x) and so on.


1. Let f : R -> R be a smooth function such that
    (f ' (x))^2 = f(x) * f ' ' (x) for all x E R.
    Suppose f(0) = 1 and f ' ' ' ' (0) = 9 , then f ' (0) is :
    A) +3 , -3          B) +(3)^1/2 , -(3)^1/2         
    C) +9 , -9          D) +(27)^1/2 , -(27)^1/2


2. A non-zero polynomial f(x) with real coefficients has the property
    that f(x)=f ' (x) * f ' ' (x) . The leading coefficient, i.e, the     
    coefficient of the highest power of x is :
    A) 1        B) 1/6        C) 3*(2)^1/2        D) 1/18


3. The non-zero real number 'k' such that f(x) = (e)^x is tangent to
    the curve at g(x) = k(x)^2 is :
    A) e        B) e^2        C) (e^2) / 4        D) (e^4) / 16


4. The real number 'a' having the property that f(a) = a is a local
    minimum of f(x) = x^4 - x^3 - x^2 + ax +1 is :
    A) 1        B) 2        C) 3        D) 4

Answer 240.
1)
f' ' ' ' (0) = 9
=> f '' ' ' (x) = 9x + a,
f ' ' (x) = (9/2)x^2 + ax + b
f ' (x) = (3/2)x^3 + ax^2/2 + bx + c
f (x) = (3/8)x^4 + ax^3/6 + bx^2/2 + cx + d


f (0) = 1 => d = 1 ... (1)


[ f ' (x) ]^2 = f (x) * f " (x)
=> [(3/2)x^3 + ax^2/2 + bx + c]^2 = [(3/8)x^4 + ax^3/6 + bx^2/2 + cx + d] * [(9/2)x^2 + ax + b]


Comparing coefficients of x^2 on both the sides,
b^2 + ac = b^2/2 + ac + (9/2) d
=> b^2/2 = 9/2 ... [because d = 1]
=> b = ± 3 ... (2)


Also,
=> [ f ' (0) ]^2 = f (0) * f " (0)
=> c^2 = b ... (3) ... [because f (0) = 1]
=> c = ± √3 ... [Only real values of c are considered]


From the above equations,
f '(0) = c = ± √3


Answer: B) +(3)^1/2 , -(3)^1/2


2)
Let f(x) = a(0)x^n + a(1)x^(n-1) + ... + a(n)
=> f '(x) = na(0)x^(n-1) + .... + a(n-1)
and f "(x) = n(n-1)a(0)x^(n-2) + .... + a(n-2)
f(x) = f '(x) * f "(x)
=> Comparing the first term on either side having the highest power of x are
a(0)x^n = n^2(n-1)a(0)x^(2n-3)
=> 2n - 3 = n => n = 3
and a(0) = n^2 (n-1) a(0)^2
=> a(0) = 1/[n^2(n-1)] = 1/[3^2*(3 - 1)] = 1/18
which is the coefficient of the highest power term of f(x)


Answer: D) 1/18.


3)
Let f(x) = e^x be the tangent to the curve g(x) = kx^2 at x = x1
=> e^x1 = kx1^2 ... (1)
The slope of the tangent at x = x1 = g'(x) at x = x1,
=> e^x1 = 2kx1 ... (2)
From (1) and (2),
kx1^2 = 2kx1
=> x1 = 0 or 2
But x1 = 0 does not satisfy (1)
=> x1 = 2
Plugging x1 = 2 in (1),
k = e^2/4


Answer C) e^2/4.


4)
f (x) = x^4 - x^3 - x^2 + ax + 1
For f (x) to be minimum, f '(x) = 0 and f "(x) > 0
f '(x) = 0
=> 4x^3 - 3x^2 - 2x + a = 0 ... (1)
Also, f (a) = a
=> a = a^4 - a^3 - a^2 + a^2 + 1
=> a^4 - a^3 - a + 1 = 0
=> a^3 (a - 1) - (a - 1) = 0
=> (a - 1) (a^3 - 1) = 0
=> (a - 1)^2 (a^2 + a + 1) = 0
=> a = 1
Plugging a = 1 in (1),
4x^3 - 3x^2 - 2x + 1 = 0
=> 4x^3 - 4x^2 + x^2 - x - x + 1 = 0
=> 4x^2 (x - 1) + x(x - 1) - (x - 1) = 0
=> (x - 1) (4x^2 + x - 1) = 0
=> x = 1
f "(x) = 12x^2 - 6x - x > 0 for x = 1
=> f (x) has a local minimum at x = 1 and the value of the local minimum is f (1) = 1 for a = 1.


Answer: A) 1.

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