The function f (x) = x^{2 }– 2 x is increasing in the interval
The function f (x) = a x + b is strict increasing for all x ∈ R iff
Since f ‘ (x) = a , therefore , f (x) is strict increasing on R iff a > 0.
Tangents to the curve x^{2}+y^{2} = 2at the points (1, 1) and (– 1, 1)
therefore , slope of tangent at (1,1) =  1 and the slope of tangent at ( 1 ,1) = 1. Hence , the two tangents in reference are at right angles.
If x be real, the minimum value of x^{2}−8x+17 is
If a differentiable function f (x) has a relative minimum at x = 0, then the function y = f (x) + a x + b has a relative minimum at x = 0 for
And y has a relative minimum at x = 0 if f ‘(0)+a = 0 . a =0.
The function f (x) = x^{2}, for all real x, is
Since f ‘(x) = 2x > 0 for x > 0,and f ‘ (x) = 2x < 0 for x < 0 ,therefore on R , f is neither increasing nor decreasing. Infact , f is strict increasing on [0 , ∞) and strict decreasing on ( ∞,0].
The function f (x) = a x + b is strict increasing for all x ∈ R if
Since f‘(x) = a , therefore , f (x) is strict increasing on R if f a > 0.
Equation of the tangent to the curve at the point (a, b) is
Hence ,slope of tangent at (a , b) = b/a . Therefore , the equation of tangent at (a , b ) is ;
a log  x  + bx^{2} + x has its extreme values at x = – 1 and x = 2, then
= 0, at x = 1 and x = 2 i.e a + 2b = 1 and 4b + a/2 = 1. ⇒ a = 2 , b = 1/2
Let f (x) be differentiable in (0, 4) and f (2) = f (3) and S = {c : 2 < c < 3, f’ (c) = 0} then
Conditions of Rolle’s Theorem are satisfied by f(x) in [2,3].Hence there exist atleast one real c in (2, 3) s.t. f ‘(c) = 0 . Therefore , the set S contains atleast one element
The function f (x) = m x + c where m, c are constants, is a strict decreasing function for all x ∈ R if
f (x) = mx + c is strict decreasing on R
if f ‘ (x) < 0 i.e. if m < 0 .
The function f (x) = x^{2}−2x is strict decreasing in the interval
f ‘ (x) = 2x – 2 = 2 (x  1) < 0 if x < 1 i.e. x x∈ (−∞,1) x∈ (−∞,1). Hence f is strict decreasing in left decreasing in (−∞,1)
The points on the curve 4 y = x^{2}−4 at which tangents are parallel to x – axis, are
Only when x = 0 and when x = 0 , y = 1. So, only at (0,1) tangent is parallel to x axis.
The maximum value of
i.e. x = e. Note that , f ‘(x) changes sign from positive to negative as we move from left to right through e. So, f (e) is maximum i.e. maximum value of f (x) is f (e)
Every continuous function is
Obviously, every differentiable function is continuous but every continuous function isn't differentiable.
The function f(x) = tan^{−1}x is
Therefore , f is strictly increasing on R.
For the curve x = t^{2}−1,y = t^{2}−t tangent is parallel to X – axis where
The slope of the normal to the curvex = a (cos θ + θ sin θ),y = a (sin θ – θ cos θ) at any point ‘θ’ is
Rolle’s Theorem is not applicable to the function f(x) =  x  for −2⩽x⩽2 because
which does not exist at x = 0 ∈ (2 , 2). So , Rolle’s theorem is not applicable.
Let f (x) = x^{3}−6x^{2}+9x+8, then f (x) is decreasing in
The equation of the normal to the curve y = sinx at (0, 0) is
therefore , slope of tangent at (0 , 0) = cos 0 = 1 and hence slope of normal at (0 , 0) is  1 .
The curve y = x^{1/5} has at (0, 0)
so,at (0 ,0) , the curve y = x1/5 has a vertical tangent.
So, f has a local minima at 2 and a local maxima at  2 .








