The instantaneous rate of change at t = 1 for the function f (t) = te−t + 9 is
The equation of the tangent to the curve y = e2x at the point (0, 1) is
Hence equation of tangent to the given curve at (0 , 1) is :
(y−1) = 2(x−0),i.e..y−1 = 2x
The smallest value of the polynomial x3−18x2+96 in the interval [0, 9] is
x3 - 18x2 + 96x = x(x2 - 18x + 96) = x[(x-9)2 + 15]
=x(x-9)2 +15 ≥ 0
Let f (x) = (x2−4)1/3, then f has a
Also, for x < 0 (slightly) , f ‘(x) < 0 and for x > 0 (slightly) f ‘(x) >0. Hence f has a local minima at x = 0 .
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph
Since the graph cuts the lines y = -1 and y = 1 , therefore ,it must cut y = 0 atleast once as the graph is a continuous curve in this case.
Let f(x) = where x>0, then f is
The equation of the tangent to the curve y=(4−x2)2/3 at x = 2 is
, which does not exist at x = 2 . However , we find that , at x = 2 . Hence , there is a vertical tangent to the given curve at x = 2 .The point on the curve corresponding to x = 2 is (2 , 0). Hence , the equation of the tangent at x = 2 is x = 2
Given that f (x) = x1/x , x>0, has the maximum value at x = e,then
The function f (x) = x3 has a
f ‘(0) = 0 , f ‘’ (0) = 0 and f ‘’’(0) = 6 . So, f has a point of inflexion at 0.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f ‘ (x) = 0 for every x, then
f ‘ (x) = 0 ⇒ f (x)is constant in (0 , 1)and also in (2, 4). But this does not mean that f (x) has the same value in both the intervals . However , if f (c) = f (d) , where c ∈ (0 , 1) and d ∈ (2, 4) then f (x) assumes the same value at all x ∈ (0 ,1) U (2, 4) and hence f is a constant function.
Let f (x) = x4 – 4x, then
The slope of the tangent to the curve x = a sin t, y = a at the point ‘t’ is
has a local maximum value = - 2 at x = - 1 and a local minimum value = 2 at x = 1.
The minimum value of (x) = sin x cos x is
Sinx cosx = 1/2 (sin2x) and minimum value of sin2x is – 1 .
In case of strict decreasing functions, slope of tangent and hence derivative is
In case of a strict decreasing function , slope of tangent and hence derivative is either negative or zero.
Let f (x) = x3−6x2+9x+18, then f (x) is strict decreasing in
Tangents to the curve y = x3 at the points (1, 1) and (– 1, – 1) are
therefore , slopes of the tangents at (1,1) and (- 1 , -1)are equal. Hence, the two tangents in reference are parallel.
Let f(x) = x25(1−x)75 for all x ∈ [0,1], then f (x) assumes its maximum value at
The stone projected vertically upwards moves under the action of gravity alone and its motion is described by x = 49 t – 4.9 t2 . It is at a maximum height when
The function f (x) = 2 – 3 x is
f (x) = 2 – 3x ⇒ f ‘ (x) = - 3 < 0 for all x ∈ R . so, f is strictly decreasing function.
Let g (x) be continuous in a neighbourhood of ‘a’ and g (a) ≠ 0. Let f be a function such that f ‘ (x) = g(x) (x−a)2 , then
Since g is continuous at a , therefore , if g (a) > 0 , then there is a nhd.of a, say (a-e , a+ e) in which g (x) is positive .This means that f ‘ (x)>0 in this nhd of a and hence f (x) is increasing at a.
The line touches the curvey =be-x/a at the point
None of the given points in (b),(c) and (d) lies on the curve y = be−x/a.
Minimum value of the function f(x) = x2+x+1 is
The function f (x) = | x | has
The modulus function has V-shaped graph,which means that it has only one minima.