f(x)=[(x^2)*∫ x→a f(t)dt]/(x-a),limx→a F(x)= Using lobida's law, ∫ x tends to a f (T) DT, what is the derivative equal to?

f(x)=[(x^2)*∫ x→a f(t)dt]/(x-a),limx→a F(x)= Using lobida's law, ∫ x tends to a f (T) DT, what is the derivative equal to?

lim(x→a)F(x)
=lim(x→a){[x ² ∫ (x→a) f(t)dt]/(x-a)
=lim(x→a)[2x∫ (x→a) f(t)dt-x ² f(x)]
=-a ² f(a)
Here, ∫ x tends to a f (T) DT according to the indefinite lower limit integral, that is, X is the lower limit and a is the upper limit

Mathematical problem.. known f '(T) = - 1, find limx → 0x / (f (t-2x) - f (t-x)) urgent!

lim(x->0) x/[f(t-2x)-f(t-x)]
={lim(x->0) [f(t-2x)-f(t-x)]/x}^(-1)
={lim(x->0) [f(t-2x)-f(t)]/x - lim(x->0) [f(t-x)-f(t)]/x}^(-1)
={lim(x->0) (-2)*[f(t-2x)-f(t)]/(-2x) - lim(x->0) -[f(t-x)-f(t)]/(-x)}^(-1)
=[-2f'(t)+f'(t)]^(-1)
=[-(-1)]^(-1)
=1

When x → x0, f (x) is infinite and limx → x0g (x) = A. from the definition, it is proved that when x → x0, f (x) + G (x) is infinite

For any M > 0, ε> 0, exists δ> 0, when | x-x0|< δ,| fx|>M,|gx-a|< ε, So | FX + Gx | > m - | a|- ε, Due to m, ε Is arbitrary, so let M1 = m - | a|- ε It is also an arbitrary number, that is, for any M1 > 0, |fx + gx| > M1, so FX + Gx is infinite

derivatives! If f '(x) = 2, Lim f (x-k) - f (x) / 2K=

F '(x) is defined as f' (x) = Lim [f (x) - f (x-k)] / K (k tends to 0)
So the answer is - 1

Let f (x) have a second-order continuous derivative and f '(x) = 0, limx-0, f' (x) / [x] = 1. Why is f (0) the minimum of F (x)?

F '(x) = 0 indicates that f (0) is the extreme value,
Limx-0 f '' (x) / [x] = 1, indicating that f '' (x) 0 can be said to be the minimum
If the second derivative of the extreme point is 0, the point is the minimum point, otherwise it is the maximum point. Because the second derivative reflects the change rate of the derivative, when the second derivative of the extreme point is 0, the derivative increases only,

Let the derivative of F (a) exist and find the limit limx close to a XF (a) - AF (x) / x-a=

Original formula = LIM (x → a) (XF (a) - AF (a) + AF (a) - AF (x)) / (x-a)
=lim(x→a)f(a)+a*(f(a)-f(x))/(x-a)
=f(a)-af'(a)