Integration: Standard Partition, Integrable over a Range

Questions on integration, see attachment.

Differentiation : Existence of Solutions

I want to prove, for the numbers a and b, that the following equation has exactly three solutions if and only if 4a^3 + 27b^2 < 0: x^3 + ax + b = 0, x in R

Real Analysis : Differentiation

If I let the function f:R->R have two derivatives with f(0)=0 and f'(x) <= f(x) for all x in R. Is f(x)=0 for all x in R?

Real Analysis: Differentiation

If I let the function f:R->R have two derivatives with f(0) = 0 and f'(x) <= f(x) for all x in R. Is f(x) = 0 for all x in R

Real Analysis : Derivatives

If I say that the function f:R->R has two derivatives, with f(0) = f'(0) = 0 and the absolute value of f"(x) is less than or equal to one, if the absolute value of x is less than or equal to 1. How can I prove that: f(x) <= 1/2 if x <= 1

Forward difference

If I be an open interval containing the point x. (x0) and suppose that the function f:I->R has two derivatives. Prove that lim as h->0 (f(x.+h) - 2f(x.) + f(x.-h))/ h^2 = f"(x.)

Simple Function Operations

1a.)Is y=x^4 a single- or multi-valued function? b.)Is y=f(x)=x^2+4x an even, odd, or neither function? c.)What is the inverse function of y=x^4 d.)What is the inverse function of (b.),y=x^2+4x? e.)Is the inverse function from (d.), odd, even, or neither?

Derivatives of Trigonometric Functions

Please see the attached file for the fully formatted problems. Find derivatives of these functions chosing appropriate methods: a) f(x) = 4x3 – 2x cos x b) g(x) = (x-2)3(x2+2x+1)3 sin2x / (x-1)2(x2+4) c) h(t) = sin (2t2 -6)

Metric Space

Let X be a metric space and x0 in X. Define a function f: X --> R (all real numbers) by f(x) = d(x,x0). Show that f is continuous. HINT: Prove the variant of the triangle inequality which says |d(x,z)-d(y,z)|< d(x,y) for any x,y,z in X

Metric Space

See atatched